CRYSTALLOGRAPHY 


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Fig.  1 


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CRYSTALLOGRAPHY 


AN   OUTLINE  OF   THE 

GEOMETRICAL    PROPERTIES 

OF   CRYSTALS 


BY 

T.  L.  WALKER,  M.A.,  PH.D. 

PROFESSOR  OF  MINERALOGY  AND  PETROGRAPHY  UNIVERSITY  OF  TORONTO 


McGRAW-HILL  BOOK  COMPANY,   INC, 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 
1914 


COPYRIGHT,  1914,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


PREFACE 

DURING  the  past  twenty  years  Dr.  Victor  Goldschmidt  of 
Heidelberg  has  done  much  to  make  crystallography  an  attractive 
subject.  His  development  of  the  two-circle  goniometer,  the 
gnomonic  projection  and  a  new  system  of  symbols  which  can  be 
read  directly  from  the  projection  has  largely  contributed  to  this 
end.  For  many  years  the  writer  has  used  in  his  classes  the 
methods  of  Goldschmidt  with  gratifying  results.  The  chief  mo- 
tive in  writing  this  book  arose  from  the  desire  to  have  in  the 
English  language  a  connected  elementary  statement  of  crystallog- 
raphy from  this  point  of  view.  It  is  hoped  that  the  present 
work  may  be  generally  useful,  not  only  as  a  presentation  of  the 
geometrical  properties  of  crystals,  but  also  on  account  of  its  fol- 
lowing in  outline  the  newer  methods  so  splendidly  developed  by 
Goldschmidt.  I  fear  I  may  be  criticised  for  my  frequent  repe- 
tition of  some  of  the  fundamental  ideas  such  as  are  usually  only 
slowly  comprehended  by  the  student.  My  defense  lies  in  the 
fact  that  it  is  neccessary  for  all  but  the  exceptional  student. 

The  extracts  from  papers  on  crystallography  contained  in  the 
last  chapter  should  serve  to  indicate  to  the  student  the  chief 
problems  of  crystallography  as  well  as  the  methods  of  solving 
them. 

The  writer  acknowledges  his  great  indebtedness  to  Dr.  Gold- 
schmidt for  assistance  in  many  ways;  to  Professors  E.  H.  Kraus, 
T.  C.  Chamberlin  and  R.  Salisbury  for  permission  to  use  figures 
contained  in  their  works;  to  colleagues  who  kindly  consented  to 
the  publication  of  extracts  of  their  papers  in  the  last  chapter 
and  to  Professor  A.  L.  Parsons  and  H.  V.  Ellsworth  for  assist- 
ance in  proof-reading  and  in  many  other  ways. 

UNIVERSITY  OF  TORONTO, 

June,  1914  T-    L-   WALKER 


CONTENTS 

PAGE 

PREFACE v 

CHAPTER  PART    I. 

I.    GENERAL  PROPERTIES. 1 

II.    FORMATION  OF  CRYSTALS 3 

From  solution 3 

By  sublimation 4 

From  fusion 4 

III.  CHEMICAL  PROPERTIES 6 

IV.  MATHEMATICAL  CHARACTERISTICS ,    .  10 

Gnomonic  projection 10 

Regular  arrangement  of  projection  points 12 

Formation  of  crystal  symbols 13 

Measurement  of  crystals 16 

(a)    The  contact  goniometer  of  Carangeot     ...  17 

(6)    Wollaston  reflecting  goniometer 18 

(c)  A  modern  reflecting  goniometer 19 

(d)  Two-circle  contact  goniometer 19 

(e)  Two-circle  reflecting  goniometer 21 

Preparation   of   gnomonic   projection   from   measure- 
ments       22 

Mathematical  determination  of  crystal  symbols  and 

constants 25 

Stereographic  projection 28 

Co-ordinate  representation  of  face  direction 29 

Derivation  of  Miller's  index  symbols 31 

General  properties  of  polyhedra   .    . 32 

V.     SYMMETRY  OF  CRYSTALS 35 

Classes  of  polyhedra  involving  only  the  following  ele- 
ments of  symmetry:   planes,  centre  and  diad,  triad, 

tetrad  and  hexad  axes 38 

Crystal  systems 39 

VI.     PHYSICAL  PROPERTIES 41 

VII.     THEORY  OF  THE  INTERNAL  STRUCTURE  OF  CRYSTALS  ....  43 


X  CONTENTS 

CHAPTER  PART    II.  PAGE 

VIII.     CUBIC  SYSTEM 47 

Symmetry  of  the  cubic  system 47 

Standard  orientation  of  cubic  crystals 47 

Axes  of  reference 48 

Classification  of  all  possible  planes  intersecting  three 

equivalent  axes  at  right  angles  to  one  another    .    .  48 

Cube      50 

Rhombic  dodecahedron      51 

Octahedron 51 

Tetrahexahedron  or  pyramidal  cube 51 

General  symbols 52 

Hexoctahedron 52 

Trisoctahedron 53 

Icositetrahedron 53 

Combinations  of  crystal  forms      54 

Limiting  forms 55 

HEMIHEDRAL  CLASSES  OF  THE  CUBIC  SYSTEM 

Plagiohedral  hemihedral  class 57 

Pentagonal  hemihedral  class 59 

Diploid      60 

Pyritohedron 61 

TETRAHEDRAL  HEMIHEDRAL  CLASS 

Tetrahedron 63 

Trigonal  tristetrahedron 64 

Tetragonal  tristetrahedron 65 

Hextetrahedron 65 

TETARTOHEDRAL  CLASS 

Projections 69 

IX.    TETRAGONAL  SYSTEM ' 70 

HOLOHEDRAL    CLASS 

Symmetry 70 

Standard  orientation 71 

Axes  of  reference 71 

Classification  of  all  planes  with  reference  to  their  inter- 
section of  the  tetragonal  axial  cross 72 

Pyramid  of  the  first  order 72 

Ditetragonal  pyramid 73 

Pyramid  of  the  second  order 74 

Basal  pinacoid 75 

Prism  of  the  first  order 76 

Prism  of  the  second  order 76 

Ditetragonal  prism 76 


CONTENTS  xi 

CHAPTER  PAGE 

IX.    TETRAGONAL  SYSTEM.     (Continued.) 

Calculation  of  the  ratio  c:  a 77 

Limiting  forms  in  the  tetragonal  system 78 

TETRAGONAL  CLASSES  OF  LOWER  SYMMETRY 

Trapezohedral  hemihedral  class 79 

Pyramidal  hemihedral  class 81 

Sphenoidal  hemihedral  class 83 

Tetartohedral  class 85 

Holohedral  hemimorphic  class 86 

Hemihedral  hemimorphic  class 86 

Projections 87 

X.    HEXAGONAL  SYSTEM 89 

Symmetry 89 

Standard  orientation 89 

Axes  of  reference 90 

Classification  of  all  possible  planes  intersecting  hex- 
agonal axes  of  reference 90 

Hexagonal  symbols 91 

HOLOHEDRAL  CLASS 

Pyramid  of  the  first  order 93 

Pyramid  of  the  second  order 93 

Dihexagonal  pyramid 94 

Hexagonal  prisms 95 

Basal  pinacoid 9g 

Limiting  forms 96 

Calculation  of  the  ratio  c :  a 96 

Projections 97 

HOLOHEDRAL  HEMIMORPHIC  CLASS 
RHOMBOHEDRAL  HEMIHEDRAL  CLASS 

Rhombohedron 99 

Scalenohedron      10Q 

TRAPEZOHEDRAL  HEMIHEDRAL  CLASS 
PYRAMIDAL  HEMIHEDRAL  CLASS 

Pyramid  of  the  third  order 102 

Prism  of  the  third  order 103 

PYRAMIDAL  HEMIHEDRAL  HEMIMORPHIC  CLASS 
TRAPEZOHEDRAL  TETARTOHEDRAL  CLASS 

Trigonal  trapezohedron 105 

Ditrigonal  prism 106 

Trigonal  pyramid 106 

Trigonal  prism 106 

RHOMBOHEDRAL  TETARTOHEDRAL  CLASS 

Rhombohedron  of  the  first  order      107 

Rhombohedron  of  the  second  order 107 


xii  CONTENTS 

CHAPTER  PAGE 

X.    HEXAGONAL  SYSTEM.      (Continued.} 

Rhombohedron  of  the  third  order 108 

Prism  of  the  third  order 108 

TRIGONAL  HEMIHEDRAL  CLASS 

Trigonal  pyramid 110 

Trigonal  prism 110 

Ditrigonal  pyramid Ill 

Ditrigonal  prism      Ill 

TRIGONAL  HEMIHEDRAL  HEMIMORPHIC  CLASS 

TRIGONAL  TETARTOHEDRAL  CLASS 
TRIGONAL  TETARTOHEDRAL  HEMIMORPHIC  CLASS 

XI.    RHOMBIC  SYSTEM •   .  114 

HOLOHEDRAL    CLASS 

Symmetry 114 

Standard  orientation 114 

Axes  of  reference 115 

Ground  form 115 

Classification  of  all  planes  with  regard  to  intersection 

of  rhombic  axes  of  reference 116 

Macropinacoid 116 

Brachypinacoid 116 

Basal  pinacoid 117 

Brachydome 117 

Macrodome 117 

Rhombic  prism 118 

Rhombic  pyramid 118 

Limiting  forms 118 

Calculation  of  the  ratio  a:  b:  c 119 

Gnomonic  projection 119 

Combinations 121 

Mineral  examples 121 

SPHENOIDAL  CLASS 
HEMIMORPHIC  CLASS 

XII.    MONOCLINIC  SYSTEM 125 

HOLOHEDRAL  CLASS 

Symmetry 125 

Standard  orientation 125 

Axes  of  reference 126 

Classification  of  all  possible  planes  with  reference  to 

intersection  of  monoclinic  axes 127 

Orthopinacoid 127 

Clinopinacoid 127 

Basal  pinacoid 127 


CONTENTS  xiii 

CHAPTER  PAGE 

XII.    MONOCLINIC  SYSTEM.     (Continued,) 

Clinodome 128 

Monoclinic  prism 128 

Negative  hemi-orthodome '  .  128 

Positive  hemi-orthodome 129 

Negative  hemi-pyramid 129 

Positive  hemi-pyramid  .        129 

Combinations 130 

Limiting  forms 130 

Calculation  of  monoclinic  ratios  a:  b:  c  and  /3  .    .    .    .  131 

Gnomonic  projection 131 

HEMIHEDRAL  CLASS 
HEMIMORPHIC  CLASS 

XIII.  TRICLINIC  SYSTEM 136 

HOLOHEDRAL    CLASS 

Symmetry 136 

Standard  orientation 136 

Axes  of  reference 137 

Classification  of  all  possible  planes  intersecting  tri- 

clinic  axes  of  reference 137 

Tetra-pyramids 138 

Hemi-macrodomes 139 

Hemi-brachydomes 139 

Hemi-prisms 139 

Pinacoids 139 

Combinations 139 

Limiting  forms 140 

Projections 140 

HEMIHEDRAL  CLASS 

XIV.  CRYSTAL  AGGREGATES 143 

PARALLEL  GROWTHS 

Irregular  aggregates 144 

Fibrous  radiating  aggregations 144 

Twinning 144 

Two  chief  types  of  twinning 146 

Repeated  twinning 147 

COMMON  TWINNING  LAWS  —  CUBIC  SYSTEM 

Spinel 148 

Fluorite 148 

Pyrite 149 

TETRAGONAL  SYSTEM 

Cassiterite 149 


xiv  CONTENTS 

CHAPTER  PAGE 

XIV.    CRYSTAL  AGGREGATES.     (Continued.) 

HEXAGONAL  SYSTEM 

Calcite 150 

Quartz 151 

RHOMBIC  SYSTEM 

Aragonite      151 

Staurolite 152 

MONOCLINIC  SYSTEM 

Gypsum 152 

Orthoclase 152 

Mica 153 

TRICLINIC  SYSTEM 

Albite 153 

General  effect  of  twinning 154 

XV.    IRREGULARITIES  OF  CRYSTAL  SURFACES 156 

Curved  faces 156 

Vicinal  planes 157 

Striations 158 

Etching  figures 158 

XVI.    CRYSTAL  DRAWINGS 160 

Preparation  of  a  plan  from  a  gnomonic  projection    .    .  161 

Clinographic  drawings 162 

XVII.    ILLUSTRATION  OF  CRYSTALLOGRAPHIC  INVESTIGATIONS  ....  165 

Cubic  system 165 

Hexagonal  system 171 

Rhombic  system 176 

Monoclinic  system 177 

Tetragonal  system 196 

Triclinic  system 196 

INDEX  .  201 


CEYSTALLOGEAPHY 

CHAPTER   I 
GENERAL   PROPERTIES 

When  a  substance  such  as  alum,  sugar  or  common  salt  is 
ground  finely  and  put  into  water  it  disappears  and  instead 
of  water  and  small  particles  of  solid  substance  we  obtain  a 
liquid  mixture  of  the  water  and  the  substance  —  a  solution. 
The  solid  has  become  liquid.  If  such  a  solution  be  set  aside 
in  an  open  vessel  so  that  the  water  may  gradually  evaporate 
we  find  that  the  solid  substance  separates  again.  This  newly 
formed  substance  on  careful  examination  will  be  found  to 
occur  in  the  form  of  numerous  little  masses  bounded  by  plane 
surfaces.  The  shape  of  the  small  masses  of  salt  is  quite 
different  from  that  of  alum  while  both  differ  in  this  respect 
from  sugar.  Those  little  masses  bounded  by  plane  surfaces 
are  known  as  crystals. 

Usually  different  substances  give  rise  to  crystals  of  different 
form  so  that  frequently  substances  otherwise  of  similar 
appearance  may  be  distinguished  from  one  another  merely 
by  a  close  examination  of  the  crystals. 

On  adding  finely  ground  powder  of  alum  and  salt  and 
sugar  to  a  beaker  of  water  so  that  the  whole  dissolves  we 
obtain  one  solution  which  contains  all  three  substances. 
Now  if  the  vessel  containing  this  solution  be  set  aside  and 
allowed  to  evaporate  gradually  the  three  substances  will 
separate  in  the  form  of  crystals  which  are  remarkable  for 
their  purity.  The  process  involved  is  known  as  crystalliza- 
tion, while  the  substance  so  separating  is  said  to  crystallize. 

While  we  are  able  to  produce  crystals  of  many  different 
kinds  by  laboratory  methods  it  is  not  only  in  this  way  that 
crystals  are  formed.  The  hard  rocks  of  the  earth's  crust 


2  CRYSTALLOGRAPHY 

are  largely  composed  of  the  crystallized  substances  known 
as  minerals.  One  of  the  commonest  of  these,  the  mineral 
quartz,  attracted  the  attention  of  the  Greek  philosophers 
many  centuries  ago.  Quartz  is  as  clear  as  water  and  fre- 
quently quite  transparent,  and  was  regarded  by  Greeks  as 
a  peculiar  form  of  ice,  possibly  resulting  from  extremely  low 
temperatures.  In  their  opinion  ice  formed  in  this  way  was 
able  to  exist  through  the  heat  of  the  summer  without  melting. 
This  substance  they  named  krystallos  —  which  means  ice. 
From  this  faulty  observation  of  the  Greeks  we  obtained  the 
word  crystal.  It  is  well  known  now  that  quartz  is  not  com- 
posed of  the  elements  common  to  ice  and  water.  It  is  an 
oxide  of  silicon  and  is  represented  by  the  chemical  formula 
SiO2. 


CHAPTER   II 
FORMATION   OF   CRYSTALS 

When  a  substance  passes  from  the  liquid  or  gaseous  state 
to  a  solid  condition  there  is  usually  a  strong  tendency  toward 
the  formation  of  crystals.  If  the  substance  be  chemically 
simple  one  kind  of  crystals  may  arise  —  if  complex,  several 
kinds.  In  the  laboratory  crystals  may  be  formed  with  no 
great  difficulty  according  to  three  distinct  methods: 

1.  From  Solution.  If  a  substance  be  dissolved  in  water 
or  any  other  liquid  and  set  aside  so  that  this  liquid  may 
evaporate,  crystals  are  gradually  formed.  The  amount  of 
a  particular  substance  which  can  be  held  in  solution  by  a 
given  solvent  varies  with  temperature.  Usually  the  higher 
the  temperature  the  greater  the  solvent  power  of  the  liquid 
in  question.  At  0°  C.  100  grammes  of  water  will  dissolve 
13.3  grammes  of  saltpetre,  while  at  100°  C.  the  same  amount 
of  water  dissolves  246  grammes.  This  is  important  in  study- 
ing the  formation  of  crystals.  As  a  result  crystals  may  be 
readily  prepared  by  setting  aside  a  hot  saturated  solution 
which  with  falling  temperature  gradually  forms  crystals  of 
the  excess  of  the  substance  which  at  the  lower  temperature 
it  is  no  longer  able  to  retain  in  the  liquid  form. 

Crystallization  from  solution  is  a  process  of  great  impor- 
tance in  the  industries,  particularly  on  account  of  the  high 
degree  of  purity  which  is  characteristic  of  the  substance 
which  has  been  crystallized  one  or  more  times.  If  a  slightly 
impure  salt  be  dissolved  and  allowed  to  crystallize  the  im- 
purity is  frequently  retained  in  the  solution,  while  the  sub- 
stance which  separates  in  the  form  of  crystals  is  of  a  higher 
degree  of  purity.  By  repeated  crystallization  the  impurity 
may  be  still  further  eliminated. 

In  nature  the  solvent  is  sometimes  exceedingly  complex 
as  in  the  magma  from  which  quartz  and  felspar  form  in 


4  CRYSTALLOGRAPHY 

the  igneous  rocks.  In  this  instance  the  temperature  of  the 
solution  .at  the  time  of  crystallization  is  above  red  heat. 
Some  of  the  substances  commonly  employed  as  solvents  are 
alcohol,  benzene,  carbon  bisulphide,  water  and  ether. 

2.  By  Sublimation.     As  a  general  rule  when  a  solid  sub- 
stance is  heated  it  is  converted  into  a  liquid  while  liquids  in 
their    turn    at  still  higher  temperatures    give   rise    to    gases. 
There  are,   however,   certain  solid  substances  which   do  not 
melt  when  heated  under  ordinary  atmospheric  conditions,  but 
give    rise   immediately   to    gases   without   first   appearing    in 
the  liquid  state.     On   the   other   hand   gases   as   a  result   of 
cooling   or   increased   pressure   normally   give   rise   to   liquids 
which  solidify  or  freeze  at  still  lower  temperatures.     To  these 
two    generalizations    there    are    some    exceptions.      A    solid 
substance,  such  as  iodine,  when  heated  under  ordinary   atmos- 
pheric pressure  passes    immediately  into  the  gaseous    condi- 
tion, and  if  this  gas  be  cooled  it  returns  directly  to  the  solid 
state   usually   in   the   form   of   crystals.     Sulphur   is   another 
substance   of    commercial    importance    which    illustrates     the 
passage  from  the  gaseous  to  the  solid  condition.     If  impure 
sulphur  be  heated  in  a  closed  chamber  it  melts  and  eventually 
becomes  a  gas.     This  gaseous  sulphur  if  led  into  a  cool  cham- 
ber becomes  immediately  solid  and  thereby  forms   myriads  of 
tiny  crystals  of  exceedingly  pure  sulphur.     A  third  example 
of  this  class  of  crystallization  is  represented  by  the  formation 
of   snow.     Warm   dry   air   readily   takes   up   water   which   is 
transported    to    various    altitudes    in    the    atmosphere.     Air 
highly   saturated   with   water   on   becoming   suddenly   cooled 
below  the  freezing  point  gives  up  part  of  the  water  in  the 
solid  form  as  crystals  of  water  or  snow.      Snow  crystals  are 
exceedingly  beautiful  and  always  characteristic  for  the  sub- 
stance from  which  they  are  formed.     They  present  the  form 
of  a  six-rayed  star  with  beautiful  plumose  shafts  (Figure  l). 

This  method  of  crystallization  may  be  easily  illustrated  by 
placing  arsenious  oxide,  ammonium  chloride  or  iodine  in  the 
lower  part  of  an  open  glass  tube  held  at  an  angle  of  30° 
and  applying  a  gentle  heat.  The  crystals  form  on  the  cooler 
part  of  the  tube. 

3.  From  Fusion.     This   process   of   crystallization   may   be 
illustrated    by    the    following    experiment.     Into    an    earthen 


FORMATION    OF    CRYSTALS  5 

crucible  place  half  a  pound  of  sulphur,  heat  until  the  whole 
mass  is  fused  and  set  aside  to  cool  until  a  firm  crust  has 
formed  on  the  surface.  Break  two  holes  in  the  crust  and 
pour  out  the  still  liquid  portion  which  occupies  the  interior 
of  the  crucible.  When  the  crucible  has  become  quite  cold, 
break  with  a  hammer  and  observe  the  growth  of  crystals 
on  the  walls  of  the  vessel.  This  experiment  may  be  per- 
formed with  equal  advantage  by  using  metallic  bismuth 
instead  of  sulphur.  In  both  cases  the  inner  lining  of  the 
crucible  will  be  found  to  be  splendidly  crystallized. 

Closely  related  to  this  type  of  crystallization  is  that  which 
may  be  observed  in  the  freezing  of  water  or  of  any  other  liquid 
composed  of  only  one  chemical  substance.  Apparently  the 
only  difference  between  the  freezing  of  water  and  the  crystal- 
lization mentioned  is  that  sulphur  and  bismuth  are  solid  at 
ordinary  temperatures  while  water  is  liquid.  It  is  necessary, 
however,  to  distinguish  sharply  between  the  crystallization 
of  sulphur  and  bismuth  from  fusion  and  the  crystallization 
of  rock  minerals  from  molten  magmas.  In  both  cases  the 
crystals  form  from  hot  liquids  but  it  is  evident  that  in  the 
separation  of  quartz,  felspar  and  mica  from  a  molten  rock 
magma,  we  are  dealing  with  phenomena  to  be  classed  with 
the  growth  of  crystals  of  alum  or  rock  salt  from  water  solu- 
tion. This  third  process  of  crystallization  from  fusion  would 
appear  to  involve  the  formation  of  crystals  from  a  simple 
chemical  individual. 


CHAPTER  III 
CHEMICAL   PROPERTIES 

As  a  general  rule  the  shape  of  the  crystal  is  characteristic 
and  more  or  less  distinctive  for  each  chemical  substance. 
By  an  observation  of  the  crystal  form  it  is  easy  to  distinguish 
many  substances  which  otherwise  have  the  same  general 
appearance.  Alum,  sugar,  magnesium  sulphate  and  common 
salt  can  be  readily  distinguished  in  this  fashion. 

Substances  which  occur  in  the  form  of  crystals  are  usually 
purer  than  the  same  substances  in  the  uncrystallized  condition. 
At  one  time  it  was  thought  that  no  two  substances  give  rise 
to  crystals  of  the  same  form  except  in  the  case  of  those  whose 
crystals  belong  to  the  cubic  system.  Mitscherlich  later  dis- 
covered that  arsenates  of  the  alkalies  were  of  the  same  crystal 
form  as  the  corresponding  phosphates.  According  to  his 
observations  the  phosphates  and  arsenates  of  sodium,  potas- 
sium and  ammonium  occur  in  crystals  of  the  same  character. 
Since  the  time  of  Mitscherlich  a  great  number  of  such  closely 
related  chemical  substances  have  been  investigated  and  it  is 
now  well  known  that  regardless  of  the  system  of  crystal- 
lization substances  chemically  analogous  frequently  produce 
crystals  which  are  geometrically  identical.  A  striking  ex- 
ample is  presented  in  the  case  of  the  alums.  Potash  alum, 
the  commonest  of  the  series,  has  the  chemical  formula  K2SO4. 
A12(SO4)3.  24  H2O.  There  is  a  large  number  of  different 
alums  such  as  might  be  obtained  by  the  substitution  of 
sodium,  ammonium,  caesium,  lithium  or  rubidium  for  potas- 
sium contained  in  the  above  formula.  Moreover  the  alumin- 
ium may  be  replaced  by  chromium,  iron  or  manganese. 
There  are  therefore  twenty-eight  different  alums  which  might 
be  obtained  by  the  simultaneous  substitution  for  both  the 
bases  contained  in  the  above  formula.  These  alums  may  be 
all  represented  by  the  formula  RI2SO4.R2m  (SO4)3.24  H2O  in 


CHEMICAL    PROPERTIES 


which  R1  stands  for  potassium,  sodium,  ammonium,  rubidium, 
caesium,  lithium  and  Rm  for  aluminium,  iron,  manganese  or 
chromium.  Such  an  expression  is  known  as  a  general  formula. 
Not  only  are  the  crystals  of  all  these  alums  identical  in  form 
but  all  these  chemical  species  may  participate  in  the  forma- 
tion of  an  individual  crystal. 

If  we  suspend  a  crystal  of  potash  alum  in  a  saturated 
solution  of  any  alum  it  will  continue  to  grow  without  change 
of  form.  As  some  of  the  alums  are  colored  the  zones  of  dif- 
ferent substances  may  be  readily  observed  by  the  variation 
in  tint.  If  several  alums  be  mixed  together  and  a  saturated 
solution  prepared  from  the  mixture  we  obtain  on  cooling  or 
evaporation  very  complex  crystals  in  which  all  the  elements 
contained  in  the  solution  are  found.  The  alums  present  to 
us,  therefore,  a  series  of  substances  of  analogous  chemical 
composition  which  crystallize  in  the  same  form.  A  crystal 
of  any  alum  suspended  in  a  saturated  solution  of  any  other 
alum  will  continue  to  grow  without  change  of  form.  Such 
a  series  of  substances  is  said  to  be  isomorphous. 

In  the  pyrite  group  of  minerals,  isomorphism  is  splendidly 
illustrated.  The  groups  of  replacing  elements  are  as  follows: 
iron,  cobalt,  nickel  and  platinum  on  the  one  hand  and  sulphur, 
arsenic,  antimony,  bismuth,  selenium  and  tellurium  on  the 
other.  The  general  formula  for  the  whole  group  is  RS2,  in 
which  R  =  Fe,  Co,  Ni  or  Pt  and  S  =  S,  As,  Sb,  Bi,  Te  or  Se. 
The  chief  minerals  of  the  pyrite  group  are  as  follows: 


Pyrite  —  FeS2 
Smaltite  —  CoAs2 
Chloanthite  —  NiAs2 
Cobaltite  —  CoAsS 
Ullmannite  —  NiSbS 
Sperrylite  —  PtAs2 


Fig.  2 


The   minerals   of   this   group   commonly   crystallize  in  forms 
represented  by  Figure  2. 

Groups  of  chemical  elements  able  to  replace  one  another 
in  isomorphous  compounds  may  be  spoken  of  as  isomorphous 
elements.  The  student  of  chemistry  will  have  little  difficulty 


8  CRYSTALLOGRAPHY 

in  selecting  examples.  Some  of  the  most  prominent  series 
of  such  elements  are  the  following: 

1.  The  alkalies  —  potassium,  sodium,  lithium,  rubidium  and 

caesium; 

2.  The    alkaline    earths  —  calcium,    strontium,    barium,    to 

which  is  frequently  added  lead; 

3.  Sulphur,    arsenic,    antimony,     bismuth,     selenium     and 

tellurium; 

4.  Phosphorus,  arsenic,  vanadium; 

5.  Iron  (FeO),  manganese  (MnO),  calcium,  magnesium  and 

zinc. 

In  view  of  the  isomorphous  replacement  of  closely  related 
elements  it  becomes  necessary  for  us  to  modify  to  a  certain 
extent  our  conclusion  as  to  the  purity  of  crystals.  This 
replacement  frequently  occurs  not  only  in  artificial  substances 
but  also  in  the  crystals  occurring  naturally  in  the  earth's 
crust.  A  mineral  containing  an  element  of  a  certain  isomor- 
phous group  will  frequently  be  found  on  careful  examination 
to  contain  at  least  small  amounts  of  some  of  the  other  ele- 
ments belonging  to  that  group.  This  may  be  illustrated  by 
the  following  analysis  of  cobaltite  (CoAsS)  from  Nordmark  in 
Norway. 

Cobalt    -  29.17%  -f-  58.7  =  .497  atomic  ratio 

Nickel  1.68%  ^  58.6  =  .028       " 

Iron        -    4.72%^  55.9  =  .084       " 

Arsenic  -  44.77%  -*-  74.9  =  .598       " 

Sulphur  -  20.23%  -^  32      =  .632       " 
For  comparison  with  the  above  the  percentage  composition 
of  the  pure  compound  may  be  indicated: 

Cobalt     -  35.41  -=-  58.7  =  .603  atomic  ratio 
Arsenic    -  45.26  -f-  74.9  =  .604       " 
Sulphur  -  19.33  -5-  32      =  .604       " 

On  examination,  these  calculations  indicate  that  in  the  case 
of  cobaltite,  the  iron  and  nickel  replace  part  of  the  cobalt  but 
always  in  proportion  to  the  atomic  weights  of  these  elements. 
Similarly  it  is  apparent  that  some  of  the  arsenic  has  been 
replaced  by  sulphur  but  here  again  these  elements  replace 
each  other  in  proportion  to  their  atomic  weights  —  74.9  parts 
by  weight  of  arsenic  are  replaced  by  32  parts  by  weight  of 
sulphur. 


.609 
1.230 


CHEMICAL    PROPERTIES  9 

Isomorphous  replacement  is  the  chief  cause  of  variation  in 
composition  in  the  case  of  crystallized  substances.  Quartz 
(SiO2)  is  exceedingly  constant  in  its  chemical  composition 
apparently  because  silicon  does  not  take  a  prominent  part 
in  any  of  the  groups  of  isomorphous  elements. 

Occasionally  a  chemical  substance  appears  in  two  or  more 
forms  of  crystals  which  are  apparently  quite  unrelated  to 
one  another.  Carbon  crystallizes  in  the  cubic  system  as 
diamond  and  also  as  graphite  in  six-sided  plates  of  the  hexag- 
onal system.  When  a  chemical  substance  appears  in  two 
such  forms  it  is  said  to  be  dimorphous;  -when  in  three  such 
forms,  trimorphous.  Titanium  dioxide  which  occurs  in  nature 
as  three  distinct  minerals  —  rutile,  anatase  and  brookite  — 
is  the  best  known  instance  of  trimorphism. 


CHAPTER   IV 
MATHEMATICAL   CHARACTERISTICS 

1.  Gnomonic  Projection.  A  line  which  is  at  right  angles 
to  a  plane  surface  is  spoken  of  as  the  normal  to  that  surface. 
If  the  direction  of  the  normal  be  known  it  follows  that  the 
direction  of  the  plane  is  also  known.  In  the  study  of  crystals, 
some  of  which  are  very  complex  polyhedra,  it  is  sometimes 
advantageous  to  represent  the  direction  of  each  plane  surface 


Fig.  3 

by  its  normal.  Figure  3  indicates  this  method  of  represent- 
ing the  direction  of  the  faces  which  occur  on  a  crystal  of 
bournonite.  Each  of  the  lines  stands  at  right  angles  to  some 
face  and  all  these  lines  are  so  placed  as  to  pass  through 
one  central  point. 

Models  representing  the  direction  of  normals  for  any 
particular  crystal  can  be  readily  prepared  by  the  use  of  sharp 
needles  which  may  be  forced  into  cork  which  serves  to  hold 
them  together  with  the  desired  direction.  Such  a  bundle  of 
needles  is  a  normal  model.  The  intersection  of  the  normals 
with  a  plane  would  be  indicated  by  a  number  of  points  on 
that  plane.  The  absolute  size  of  the  area  over  which  these 
points  would  be  distributed  depends  upon  the  distance  be- 


MATHEMATICAL  CHARACTERISTICS 


11 


tween  the  plane  and  the  centre  of  the  crystal.  The  scale  is 
proportional  to  this  distance.  Such  a  plane  of  points  is 
called  a  gnomonic  projection.  In  the  projection  the  direc- 


Fig.  4 


tions  of  all  the  faces  of  a  crystal  may  be  accurately  indicated. 

The  amount  of  information  which  is  conveyed  by  a  gnomonic 

projection    is    more    or    less    dependent    upon    its    direction. 

Ordinarily   the   plane   of   projection   is    horizontal   and   since 

crystals   are   always   set  up  or   oriented  in 

a  particular  way  for  study  it  follows  that 

such  a  projection  is  at  right  angles  to  the 

accepted  vertical  direction   of   the    crystal. 

Figure   4    represents   a    projection    parallel 

to  the    base    of  the  topaz  crystal  shown  in 

Figure  5.     The  distance  between  the  centre 

of  the  crystal  and  the  plane  of  projection 

is   indicated  by  the  radius   of   the  circle   in 

the  projection.     For  convenience  this  height 

is  ordinarily  five  centimeters.  Fig.  5 


12  CRYSTALLOGRAPHY 

2.  Regular  Arrangement  of  Projection  Points.  A  brief 
examination  of  this  projection  suffices  to  show  that  the  points 
are  distributed  in  a  regular  manner.  They  show  a  marked 
tendency  to  occur  in  lines.  Where  three  points  occur  on  a 
straight  line  it  will  be  found  that  these  points  correspond  to 
three  faces  on  the  crystal  which  meet  in  parallel  edges  or 
which  would  form  parallel  edges  if  extended.  A  series  of 
faces  on  a  crystal  intersecting  in  parallel  edges  is  called 
a  zone  and  a  series  of  points  lying  in  a  straight  line  in 
a  gnomonic  projection  may  be  referred  to  as  a  zone  line. 
In  this  projection  of  topaz  many  such  zone  lines  occur. 
Some  of  these  lines  are  parallel  to  other  lines.  In  the 
topaz  projection  there  are  two  sets  of  parallel  lines  at 
right  angles  to  one  another.  If  a  projection  be  turned 
so  that  one  set  of  parallel  lines  is  right  and  left  then 
the  other  will  appear  fore  and  aft.  Let  us  examine  the 
position  of  those  parallel  lines  having  the  right  and  left 
direction.  One  of  them  passes  through  the  centre  of  the 
picture;  others  before  or  behind  this  central  line.  If  the  dis- 
tance between  any  pair  of  these  right  and  left  lines  be  meas- 
ured it  will  be  found  that  it  is  rationally  related  to  the 
distance  between  any  other  pair.  Similar  relations  obtain 
in  connection  with  the  fore  and  aft  zone  lines.  While  the 
distances  between  pairs  of  right  and  left  lines  are  simply 
related  to  one  another  there  is  no  simple  relationship  between 
these  distances  and  the  corresponding  distances  between  the 
fore  and  aft  lines.  For  topaz  these  two  sets  of  measurements 
are  always  irrationally  related  to  one  another.  Set  a  pair 
of  compasses  so  as  to  mark  exactly  the  distance  between 
any  pair  of  right  and  left  lines  and  then  measure  with  this 
unit  the  distance  of  each  projection  point  from  the  central 
right  and  left  line.  The  number  of  paces  will  be  found  to  be 
either  a  small  whole  number,  a  simple  fraction,  infinity  or 
zero.  Similar  relationships  hold  with  regard  to  these  points 
when  measured  from  the  central  fore  and  aft  line.  As  has 
been  already  remarked  in  the  case  of  topaz  the  length  of  the 
fore  and  aft  pace  is  irrationally  related  to  the  right  and  left 
pace.  Some  crystals  are  very  complex  on  account  of  the 
number  of  faces  which  occur  upon  them  but  in  all  cases  only 
such  faces  occur  as  are  in  accord  with  the  rationality  of  the 


MATHEMATICAL  CHARACTERISTICS  13 

pace  numbers.  Faces  having  directions  which  would  give 
us  projection  points  out  of  harmony  with  this  generalization 
do  not  occur.  This  first  generalization  may  be  referred  to  as 
the  law  of  rationality  of  the  pace  numbers.  Sometimes  it 
is  called  the  law  of  the  rationality  of  the  axial  parameters. 
The  propriety  of  this  name  will  be  apparent  later.  This  is 
the  first  and  most  remarkable  law  of  crystallography. 

3.  Formation  of  Crystal  Symbols.  In  dealing  with  such 
complex  polyhedra  it  is  not  only  desirable  but  necessary  to 
devise  some  system  by  which  the  multitude  of  faces  can  be 
accurately  and  simply  indicated.  Many  systems  of  symbols 
have  been  suggested.  One  of  these  can  be  derived  immedi- 
ately from  the  gnomonic  projection.  Having  already  arbi- 
trarily selected  fore  and  aft  and  right  and  left  pace  lengths 
.let  us  determine  the  number  of  such  paces  forward  or  back- 
ward involved  in  any  projection  point.  If  the  point  lie  in 
front  of  the  right  and  left  medial  line  the  number  may  be 
indicated  as  positive;  if  behind,  negative.  Next  let  us 
determine  the  pace  numbers  in  the  same  fashion  in  regard  to 
the  fore  and  aft  medial  line.  Measurements  to  the  right  are 
positive;  measurements  to  the  left,  negative.  Let  it  be  agreed 
that  the  symbol  for  any  face  consists  of  the  fore  and  aft 
pace  number  with  its  appropriate  sign,  followed  by  the  right 
and  left  pace  number  similarly  indicated.  These  projection 
symbols  were  first  suggested  by  Prof.  Victor  Goldschmidt 
and  are  commonly  known  as  the  Goldschmidt  symbols. 
Goldschmidt  has  introduced  a  further  abbreviation  in  that 
when  the  two  pace  numbers  are  identical  the  second  is  sup- 
pressed, thus  00  =  0,  oo  oo  =  oo,  11  =  1,  33  =  3.  The 
system  of  symbols  popularized  by  Professor  Miller  and  com- 
monly known  by  his  name  can  be  readily  obtained  from  the 
projection  symbols  by  placing  the  figure  1  after  the  projec- 
tion symbol  and  then  simplifying  by  dividing  or  multiplying 
so  as  to  get  rid  of  fractions  and  °° .  As  Miller's  system  has 
been  accepted  in  all  parts  of  the  world  it  is  desirable  to  be 
able  readily  to  transform  any  other  system  which  may  be 
employed  into  this  international  system.  In  the  descriptive 
part  of  this  book  it  is  proposed  to  regularly  use  both  the 
systems  of  symbols  outlined.  The  derivation  of  symbols  from 
the  gnomonic  projection  is  not  always  so  simple  as  it  is  in  the 


14  CRYSTALLOGRAPHY 

case  of  topaz,  but  on  the  whole  comparatively  little  effort 
is  required  to  obtain  a  fair  facility  in  this  short  hand  descrip- 
tive method. 

On  the  projection  of  topaz  the  points  are  so  distributed  as 
to  be  symmetrical  about  the  two  medial  lines  which  stand  at 
right  angles  to  one  another.  If  a  normal  be  dropped  from 
point  11  to  the  fore  and  aft  line  and  produced  an  equal  dis- 
tance beyond  the  point  of  intersection  we  observe  that  it 
reaches  the  point  11.  These  points  are  said  to  be  symmetri- 
cally arranged  with  regard  to  the  fore  and  aft  line.  The 
points  11  and  11  are  also  symmetrically  placed  with  regard  to 
the  right  and  left  line.  It  may  be  noted  that  the  occurrence 
of  a  point  which  is  not  contained  in  either  of  the  lines  of  symmetry 
necessarily  implies  the  occurrence  of  three  other  points  —  the 
symmetry  about  two  lines  at  right  angles  to  one  another  requires 
this.  On  the  other  hand,  a  projection  point  which  lies  on  one 
of  these  lines  of  symmetry  will  necessarily  be  accompanied 
by  a  second  point,  the  two  being  placed  symmetrically  about 
the  other  line.  The  third  possible  condition  consists  in  the 
occurrence  of  the  projection  point  at  the  intersection  of  the 
two  lines  of  symmetry.  Such  a  point  may  occur  alone.  It 
appears,  therefore,  that  on  crystals  there  are  certain  sets 
of  faces  all  of  which  may  be  expected  when  any  one  of  them 
occurs.  In  the  projection  in  question  the  occurrence  of  the 
face  11  requires  the  simultaneous  occurrence  of  11,  11,  and  11. 
Such  sets  of  faces  are  known  as  crystal  forms.  As  indicated 
in  the  topaz  projection  there  are  certain  forms  which  are 
there  represented  by  four  faces,  others  by  two  and  one  by  one. 

There  is  another  element  of  symmetry  well  represented  in 
the  topaz  projection.  If  a  line  be  drawn  from  any  projec- 
tion point  to  the  point  where  the  two  lines  of  symmetry 
meet,  and  produced  an  equal  distance  beyond  it  will  always 
terminate  in  a  similarly  situated  point.  This  peculiarity  of 
the  projection  is  said  to  be  due  to  its  symmetry  about  its 
central  point.  The  degree  of  symmetry  as  exhibited  in  pro- 
jections varies  with  that  of  the  crystal.  The  topaz  projec- 
tion is  symmetrical  about  two  lines  at  right  angles  to  each 
other  and  also  about  its  central  point.  This  arrangement 
of  projection  points  with  reference  to  certain  lines  and  points 
of  symmetry  is  so  regular  that  it  has  been  formulated  as 


MATHEMATICAL  CHARACTERISTICS  15 

the  law  of  symmetry.  Crystal  faces  occur  in  sets  of  such 
numbers  and  with  such  inclinations  as  to  preserve  the  sym- 
metrical arrangement  with  regard  to  the  point,  lines  and  planes 
of  symmetry  which  are  characteristic  for  the  crystal  individual  in 
question. 

There  are  many  substances  which  crystallize  in  accordance 
with  the  same  symmetry  exhibited  by  topaz,  but  if  we  were 
to  classify  in  accordance  with  symmetry  all  the  crystal  indi- 
viduals which  have  been  carefully  measured  and  examined, 
it  would  be  necessary  for  us  to  provide  about  thirty  distinct 
classes.  Some  substances  crystallize  so  as  to  exhibit  a  large 
number  of  planes  of  symmetry.  Generally  the  higher  the 
number  of  planes  of  symmetry  the  larger  the  number  of  faces 
comprising  a  crystal  form.  The  diamond  possesses  a  very 
high  degree  of  symmetry  and  occasionally  its  crystals  present 
to  us  forms  involving  forty-eight  faces.  As  will  be  seen  later 
the  classification  of  crystals  is  based  upon  symmetry. 

The  distance  measured  on  this  gnomonic  projection,  from 
the  central  point  to  the  various  projection  points  belonging 
to  any  one  set,  is  constant.  This  is  due  to  the  fact  that  the 
angles  of  the  crystal  formed  by  these  several  pairs  of  faces 
are  identical.  This  would  be  true  without  regard  to  the  size 
of  the  crystal,  the  locality  from  which  it  is  obtained  or  even 
the  relative  sizes  of  the  faces.  This  regularity  which  is  so 
plainly  shown  in  the  gnomonic  projection  as  to  be  scarcely 
worthy  of  mention  was  observed  in  1669  by  Nicholas  Steno 
who  first  formulated  for  us  the  law  of  the  constancy  of  inter- 
facial  angles.  Under  given  physical  conditions  and  for  a 
particular  crystal  individual  the  angle  between  corresponding 
pairs  of  faces  is  constant. 

Before  assigning  symbols  to  the  projection  points  we  found 
it  necessary  to  arbitrarily  select  fore  and  aft  and  right  and 
left  paces.  The  ratios  existing  between  the  fore  and  aft 
pace,  the  right  and  left  pace  and  the  height  of  the  plane  of 
projection  above  the  centre  are  constant  and  characteristic 
for  the  mineral  topaz.  For  this  type  of  crystal  these  three 
values  are  irrationally  related  to  each  other.  There  are 
many  other  substances  presenting  the  same  symmetry  as 
topaz  for  which  similar  projections  might  be  formed,  but  it 
is  very  improbable  that  for  any  other  substance  would  the 


16  CRYSTALLOGRAPHY 

ratio  between  these  three  values  even  closely  approximate 
the  ratios  shown  by  topaz.  As  it  will  be  necessary  to  refer 
frequently  to  these  ratios  let  us  indicate  for  the  sake  of 
brevity  the  fore  and  aft  pace  as  p0,  the  right  and  left  pace 
as  qo  and  the  height  of  the  plane  of  projection  as  r0.  The 
ratio  p0:  qo:  r0  for  a  substance  such  as  topaz  is  as  charac- 
teristic for  it  as  its  specific  gravity,  its  index  of  refraction  or 
its  coefficient  of  expansion.  These  ratios  are  called  crystal- 
lographic  constants.  Later  we  must  study  the  methods  in- 
volved in  the  exact  mathematical  determination  of  these 
ratios.  For  the  present  po  may  be  found  by  measuring 
carefully  from  point  to  point  forward  and  backward.  The 
same  method  may  be  applied  to  q0.  If  the  projection  were 
prepared  at  a  height  of  five  centimetres  above  the  centre  of 
the  crystal,  then  r0  is  equal  to  five  centimetres.  For  topaz 
the  values  obtained  from  these  measurements  should  be  as 
follows:  p0:  qo^  r0  :  :  9.0245:4.7695:5.  Now  if  we  simplify 
making  r0  unity  the  ratio  becomes  p0:  qo:  r0  :  :  1.8049:  .9539:  1. 
This  is  a  graphic  method  for  determining  from  the  gnomonic 
projection  the  approximate  values  of  the  crystallographic 
constants. 

The  individuality  of  crystallographic  constants  may  be 
readily  observed  by  an  examination  of  the  following  table  in 
which  the  characteristic  values  are  indicated  for  several  sub- 
stances having  the  same  symmetry  as  topaz: 

Po  qo  r0 

Anglesite  1.6421  1.2894  1 

Anhydrite  1.1204  1.0008  1 

Cerussite  1.1853  .7230  1 

Celestite  1.6426  1.2830  1 

Topaz  1.8049  .9539  1 

The  graphic  determination  must  not  be  regarded  as  merely 
suitable  for  purposes  of  demonstration.  From  a  carefully 
prepared  gnomonic  projection  values  may  be  obtained  always 
accurate  to  the  first  three  decimals.  Naturally  the  most 
accurate  results  are  to  be  obtained  by  mathematical  calculation. 

4.  Measurement  of  Crystals.  The  preparation  of  a  gno- 
monic projection  implies  very  accurate  measurement  of  the 
crystal  to  be  represented.  Various  instruments  known  as 


MATHEMATICAL  CHARACTERISTICS 


17 


goniometers    have    been    devised    for    crystal    measurement. 

The  earlier  instruments  were  of  simple  construction,   easily 

operated,    and   correspondingly   inaccurate.     During  the   last 

one  hundred  and  thirty  years  many  improvements  have  been 

made  so  that  it  is  now  possible  to  measure  the  best  crystals 

with  an  accuracy  which  permits  variations  of  only  a  fraction 

of   a   minute.     The   instruments   of   the   earlier   period   were 

scarcely  designed  for  an  accuracy  greater  than  one-half  degree. 

These  goniometers  fall  into  two  classes.     The  earlier  instru- 

ments are  known  as  contact  goniometers  and  were  particu- 

larly useful  for  the  examination  of  large  crystals  regardless 

of  the  roughness   or 

smoothness    of    the 

crystal   face.     The 

more  modern  instru- 

ments are  known  as 

reflecting    goniomet- 

ers and  give  good  re- 

sults    with     crystals 

half  a   millimetre  in 

diameter,  provided 

the  faces  are  bright 

so  as  to  reflect  light. 

In  the  following  para- 

graphs some  of    the 

chief  types  are  briefly 

sketched. 


Fig.  6 


(a)  The  Contact  Goniometer  of  Carangeot.  The  first  instru- 
ment for  the  measurement  of  solid  angles  was  devised  by 
Carangeot  in  Paris  in  1780  and  was  used  by  him  in  the  mak- 
ing of  crystal  models  from  clay.  It  consists  of  a  graduated 
semicircle  to  the  centre  of  whose  diameter  is  attached  a  swing- 
ing bar.  A  crystal  whose  angle  is  to  be  measured  is  carefully 
adjusted  between  the  two  bars.  The  distant  end  of  the 
movable  bar  indicates  the  magnitude  of  the  crystal  angle 
on  the  graduated  semicircle.  It  is  necessary  in  making  such 
measurements  to  be  careful  that  the  lines  of  contact  for  the 
two  bars  on  the  crystal  surfaces  are  at  right  angles  to  the 
edge  between  the  faces.  If  this  be  not  kept  in  mind  very 
variable  angles  may  be  obtained.  When  we  say  that  the 


18 


CRYSTALLOGRAPHY 


angle  between  a  pair  of  faces  is  120°  we  mean  that  a  pair 
of  lines,  drawn  from  a  point  on  the  edge  between  them  and 
at  right  angles  to  the  edge,  form  an  angle  of  120°  with  one 
another.  The  goniometer  of  Carangeot  is  in  use  at  the  pres- 
ent day  and  has  undergone  very  little  modification. 

(6)  Wollaston  Reflecting  Goniometer.  The  first  reflecting  goni- 
ometer was  prepared  by  Wollaston  in  1809  and  consisted  of  a 
graduated  circle  movable  about  a  horizontal  axis.  A  crystal 
whose  interfacial  angle  is  to  be  measured  is  attached  to  this 

axis  in  such  a  way  that  the 
edge  between  the  faces  forming 
the  angle  is  parallel  to  the  axis. 
The  crystal  faces  must  be  bril- 
liant reflectors.  In  a  dark  room 
a  light  is  placed  so  that  the  re- 
flection from  one  of  the  faces 
reaches  the  eye  of  the  observer. 
If  the  disc  be  revolved  the  re- 
flection from  the  other  face  will 
not  reach  the  eye  until  the 
second  face  as  a  result  of  the 
revolution  assumes  a  position 
parallel  to  that  of  the  origi- 
nal reflecting  face.  The  angle 
through  which  the  crystal  must 
be  revolved  will  represent  the 
true  interfacial  angle,  providing 
the  plane  including  the  incident 


Fig.  7 


and  reflected  rays  is  at  right  angles  to  the  edge  of  the  crystal. 
With  the  earlier  reflecting  goniometers  it  was  difficult  to  be 
sure  that  this  plane  was  always  exactly  at  right  angles  to 
the  edge  of  the  crystal. 

By  the  use  of  the  old  style  con- 
tact goniometer  the  inner  angle  was 
measured  so  that  on  a  six-sided  prism 
the  angle  would  be  spoken  of  as  120°. 
With  the  reflecting  goniometer  it  is 
not  necessary  to  turn  the  crystal 
through  120°  to  secure  a  reflection 
from  the  second  face  but  only  through  Fig.  8 


MATHEMATICAL  CHARACTERISTICS 


19 


60°  which  is  the  supplement  of  the  angle  as  observed  when 
measurement  has  been  made  with  the  contact  goniometer. 
These  angles  are  known  as  inner  and  outer  angles.  The 
propriety  of  these  terms  is  illustrated  by  Figure  8. 

If  the  light  be  just  reflected  from  the  face  AB  and  the 
crystal  be  turned  till  BC  attains  the  direction  at  first  pos- 
sessed by  AB,  the  second  reflection  will  occur.  The  angle 
through  which  the  crystal  must  be  turned  is  represented  in 
the  figure  EDF,  which  is  equal  to  the  outer  angle  CBG.  .  In 
the  following  pages  the  interfacial  angles  refer  —  unless  other- 
wise specially  mentioned  —  to  the  outer  angles. 

(c)  A  Modern  Reflecting  Goniometer.  In  this  instrument  the 
graduated  circle  is  horizontal  and  the  instrument  is  supplied 


Fig.  9 

with  a  collimator  through  which  the  light  is  led  to  the  crystal, 
while  a  telescope  carries  the  reflected  ray  from  the  crystal 
to  the  eye.  The  crystal  is  supported  on  a  bar  which  rises 
from  the  centre  of  the  graduated  disc  and  must  be  so  placed 
that  the  reflecting  face  lies  at  the  point  where  the  axes  of  the 


20 


CRYSTALLOGRAPHY 


collimator,  telescope  and  graduated  circle  meet.  Moreover 
it  is  necessary  to  adjust  or  orient  the  crystal  so  that  the 
edge  over  which  the  angle  is  to  be  measured  is  normal  to 
the  plane  containing  the  axes  of  the  collimator  and  the  tele- 
scope. When  this  adjustment  has  been  made  the  inner  angle 
may  be  determined  by  bringing  these  two  faces  in  succession 
to  the  point  of  reflection,  reading  the  graduated  circle  for 
each  position  and  noting  the  difference.  This  instrument, 
like  all  reflecting  goniometers,  must  be  operated  in  a  dark 
room  with  a  special  source  of  light  placed  in  line  with  a 
collimator.  This  instrument  is  a  modern  example  of  a  one-circle 
reflecting  goniometer.  Many  mechanical  attachments  are 
introduced  to  facilitate  the  exact  orientation  of  the  crystal 
and  to  secure  such  a  reflection  as  to  render  accurate  reading 
possible  (Figure  9). 

(d)  Two-circle  Contact  Goniometer.  This  instrument  con- 
sists of  a  horizontal  graduated  circle  from  whose  centre 
rises  a  rod  to  which  the  crystal  may  be  cemented.  A  verti- 
cal graduated  circle  having  the  crystal  approximately  at  its 
centre  is  also  provided.  The  crystal  to  be  measured  is  at- 
tached to  the  rod  with 
the  ordinary  standard 
orientation  and  then  by 
means  of  rotation  of  the 
horizontal  disc  and  by 
adjustment  of  the 
sliding  rod  which  serves 
as  a  radius  of  the  verti- 
cal circle  each  crystal 
face  in  turn  is  brought 
into  such  a  position  that 
its  normal  coincides  in 
direction  with  the  mov- 
able rod,  that  is,  the 
small  circular  disc  on  the 
end  of  this  radial  rod 
is  parallel  to  the  crystal 
face.  The  vertical  cir- 
cle is  so  graduated  that 


Fig.  10 


the  zero  is  at  its  highest  point  and  from  that  zero  point  the 


MATHEMATICAL  CHARACTERISTICS  21 

numbers  increase  both  ways  to  a  maximum  of  90°.  This 
is  a  simple  style  of  two-circle  contact  goniometer.  There 
are  two  angles  to  be  read  for  each  crystal  face.  By  this 
method  is  obtained  a  record  of  the  direction  of  each  face 
normal.  One  set  of  measurements  corresponds  in  a  way  to 
longitude.  These  are  derived  from  the  reading  of  the  hori- 
zontal circle  which  on  this  instrument  reads  from  zero  to  360° 
while  longitude  ordinarily  numbers  in  both  directions  and 
attains  a  maximum  of  180°.  A  second  set  of  measurements 
corresponds  to  latitude  but  again  in  a  reverse  order.  The 
latitude  of  the  pole  on  an  earth  globe  is  90°,  the  latitude 
of  the  equator  being  zero.  In  the  reading  derived  from  this 
two-circle  goniometer  the  latitude  of  the  pole  is  zero  and  of 
the  equatorial  direction  90°.  This  instrument  (Figure  10) 
and  the  one  about  to  be  described  were  devised  by  Prof. 
Victor  Goldschmidt  and  are  among  the  most  suitable  gonio- 
meters for  didactic  purposes.* 

(e)  Two-circle  Reflecting  Goniometer.  The  instrument  rep- 
resented in  Figure  11  consists  of  two  graduated  circles,  one 
of  which  is  horizontal  and  parallel  to  the  plane  of  the  colli- 
mator  and  the  telescope.  The  second  is  borne  by  an  arm 
in  such  a  way  that  it  is  vertical  and  revolves  about  a  hori- 
zontal axis,  lying  in  the  plane  of  the  collimator  and  telescope, 
while  the  axes  of  the  collimator,  telescope,  horizontal  and 
vertical  circles  meet  in  one  central  point.  The  arm  which  car- 
ries the  vertical  circle  swings  freely  about  the  axis  of  the  hori- 
zontal circle.  An  arm  similar  to  that  which  carries  the  vertical 
circle  supports  the  telescope,  which  is  clamped  when  the  instru- 
ment is  being  used  for  crystal  measurement.  The  crystal 
to  be  measured  is  attached  to  the  axis  of  the  vertical  circle 
in  such  a  manner  that  the  standard  vertical  direction  of 
the  crystal  coincides  with  the  direction  of  this  axis.  For 
measurement  it  is  necessary  that  the  crystals  should  be 
centred  so  that  the  reflecting  face  is  very  near  the  point  of 
junction  for  the  axes  of  the  telescope,  the  collimator  and 
the  vertical  circle.  This  instrument  is  in  many  respects 
similar  to  the  last.  In  measurement  we  get  two  sets  of 

*  The  instruments  represented  in  figures  10  and  11  are  manufactured  by 
Fritz  Rheinheimer,  Landfriedstr.  6,  Heidelberg,  Germany. 


22 


CRYSTALLOGRAPHY 


angles,  the  readings  from  the  horizontal  circle  coincide  with 
those  from  the  vertical  circle  of  the  contact  goniometer  and 
the  readings  from  the  vertical  circle  correspond  with  those 
from  the  horizontal  circle  of  the  contact  goniometer.  Very 
accurate  measurements  can  be  made  with  this  instrument. 
It  has  a  great  advantage  over  the  ordinary  one-circle  reflect- 
ing goniometer  in  that  it  is  possible  to  read  with  one  setting 
up  of  the  crystal  all  the  faces  which  are  ordinarily  necessary. 
With  a  one-circle  reflecting  goniometer  it  is  necessary  to 
orient  afresh  the  crystal  for  each  zone  of  faces  to  be  measured. 
In  the  case  of  the  very  small  crystals  this  frequent  orienta- 
tion may  become  very  tedious. 


Fig.  11 

5.  Preparation  of  Gnomonic  Projection  from  Measure- 
ments. The  measurements  obtained  by  the  use  of  the  two- 
circle  goniometer  imply  first  a  polar  direction  corresponding 
in  a  way  with  the  pole  of  the  earth  as  represented  in  the  earth 
globe  and  second,  a  plane  of  first  meridian  or  a  plane  of 
reference  to  which  are  referred  those  angles  corresponding  to 
longitude.  In  the  case  of  crystals  the  pole  coincides  with 
the  standard  vertical  direction  of  the  crystal  while  the  plane 
of  the  first  meridian  as  shown  by  the  measurements  is  merely 


MATHEMATICAL  CHARACTERISTICS  23 

a  chance  direction.  It  is  in  fact  that  plane  through  the 
crystal  which  contains  the  axis  of  the  vertical  circle  and  the 
zero  point  upon  that  circle.  Starting  with  these  measure- 
ments the  present  problem  is  the  preparation  of  a  gnomonic 
projection.  Let  the  plane  of  projection  be  at  right  angles  to 
the  pole,  which  coincides  with  the  standard  vertical  direc- 
tion of  the  crystal,  and  five  centimetres  from  the  central 
point  of  the  crystal  where  all  the  normals  may  be  supposed 
to  intersect.  It  is  equally  easy  to  prepare  such  a  projection 
at  any  other  distance  from  the  centre,  but  if  the  distance  be 
less  than  five  centimetres  the  points  are  so  close  together 
that  any  error  in  plotting  is  relatively  more  serious,  while  if 
the  distance  from  the  centre  be  greater  the  projection  becomes 
unnecessarily  large  without  securing  any  compensating  advan- 
tage. In  attempting  to  make  such  a  projection  we  should 
be  provided  with  drafting  paper  on  which  two  lines  have 
been  drawn  at  right  angles  to  one  another.  On  paper  spe- 
cially prepared  for  such  projections  a  network  of  lines  parallel 
to  these  two  are  arranged  at  intervals  of  one  centimetre 
with  further  subdivision  in  lighter  lines  one  millimetre  apart. 
Enclosing  this  whole  netted  area  is  a  graduated  circle  whose 
centre  lies  at  the  intersection  point  of  two  of  the  heavier 
lines.  This  circle  is  graduated  from  zero  to  360°,  ihe  numbers 
increasing  in  the  direction  of  the  hands  of  a  clock.  The 
zero  point  for  this  graduation  is  at  the  right  end  of  the  right 
and  left  line  passing  through  the  centre.  Various  sheets 
of  paper  intended  for  this  purpose  B  c 

are  prepared  and  may  be  purchased  A 
with  all  the  preliminary  lines  upon 
them  so  that  it  is  not  necessary  to 
spend  time  upon  preparing  the  paper 
prior  to  making  the  projection. 
With  such  a  sheet  of  paper  the  prep- 
aration of  a  gnomonic  projection 
from  measurements  is  very  simple.  jrjg  12 

Take  a  particular  face  whose  readings 

are  31°  30'  on  the  horizontal  circle  (corresponding  to  latitude) 
and  217°  20'  on  the  vertical  circle  (corresponding  to  longitude). 
The  pole  of  the  projection  is  marked  by  the  point  at  the 
centre  of  the  circle.  The  projection  point  for  this  face  lies 


24  CRYSTALLOGRAPHY 

on  a  line  connecting  the  centre  with  the  reading  217°  20' 
on  the  projection  sheet.  To  find  its  position  on  that  line 
it  is  only  necessary  to  measure  a  length  from  the  centre 
equal  to  five  times  the  tangent  of  the  angle  of  the  horizontal 
circle  in  centimetres.  This  angle  corresponding  to  latitude 
is  ordinarily  designated  by  the  letter  p.  The  diagram  (Fig- 
ure 12)  represents  a  section  through  the  plane  of  projection, 
the  centre  of  the  crystal  and  the  projection  point  for  the 
face  in  question.  It  is  readily  apparent  from  an  examina- 
tion of  this  diagram  that  the  distance  of  a  projection  point 
from  the  pole  is  represented  by  the  product  of  the  tangent 
of  p  and  the  height  of  the  plane  of  projection  above  the 
centre  of  the  crystal. 

E  =  crystal  centre 

AD  =  plane  of  projection 

B  =  projection  point 

ZBEC  =  p 

CE  =  h  =  height  of  plane  above  centre 

BC  =  distance  of  projection  point  from  pole 

BC  BC 

— —  =  tan  p,  also  — —  =  tan  p 
CrL  h 

.'.  BC  =  tan  p  .  h. 

By  this  method  the  various  projection  points  may  be 
located  and  the  projection  completed.  Where  the  value  of 
P  exceeds  80°  the  distance  of  the  projection  point  from  the 
centre  may  be  so  great  as  to  fall  outside  the  bounds  of  the 
paper.  Such  cases  are  unusual  except  where  p  is  90°.  In 
this  case  the  intersection  of  the  plane  by  a  normal  with  p  = 
90°  will  only  occur  at  <»  so  that  it  is  usual  to  indicate  projec- 
tion points  of  this  class  of  face  by  means  of  an  arrow  on  the 
border  of  the  sheet  of  paper,  the  arrow  lying  on  the  line 
joining  the  centre  with  the  projection  point. 

If  a  projection  prepared  in  this  way  be  examined  as  to  the 
direction  of  its  lines  of  symmetry  it  will  usually  be  found  that 
none  of  these  lines  have  the  right  and  left  and  fore  and  aft 
direction  but  that  it  is  necessary  to  revolve  the  projection 
through  a  certain  angle  in  order  to  secure  this  arrangement. 
As  has  already  been  stated  there  is  a  standard  vertical  direc- 
tion arbitrarily  agreed  upon  for  each  type  of  crystal.  The 


MATHEMATICAL  CHARACTERISTICS  25 

standard  orientation  for  each  crystal  also  prescribes  right 
and  left  and  fore  and  aft  directions.  In  the  case  of  the 
rhombic  system,  to  which  topaz  belongs,  one  of  the  lines  of 
symmetry  must  be  right  and  left  and  the  other  fore  and  aft. 
A  simple  calculation,  which  need  not  be  outlined  here,  is 
necessary  to  determine  the  number  to  be  subtracted  from 
the  readings  obtained  on  the  vertical  circle  so  as  to  give  the 
crystal  its  standard  orientation  in  the  projection.  This  angle 
through  which  the  projection  must  be  revolved  or  which 
must  be  subtracted  from  or  added  to  the  readings  of  the 
vertical  circle  is  known  as  v0  according  to  Goldschmidt. 

Figure  4  represents  a  gnomonic  projection  for  topaz  on 
which  the  forms  present  on  Figure  5  are  indicated. 

6.  Mathematical  Determination  of  Crystal  Symbols  and 
Constants.  The  method  to  be  pursued  in  the  graphic  deter- 
mination of  symbols  and  constants  has  been  already  indicated. 
As  will  be  readily  apparent  the  graphic  method  does  not 
give  the  most  accurate  results.  There  are  two  sources  of 
error.  First,  the  projection  point  may  be  placed  a  short 
distance  from  its  theoretical  position.  Second,  when  we 
attempt  to  measure  the  distance  between  points  it  is  difficult 
to  read  distances  less  than  one-half  millimetre.  More  accurate 
results  may  be  obtained  from  the  crystal  measurements  by 
mathematical  calculation.  The  projection  of  topaz  falls  into 
four  quadrants.  In  each  quadrant  similar  sets  of  points  are 
present.  This  is  in  accordance  with  symmetry  about  two 
lines  at  right  angles  to  one  another.  Before  proceeding  to 
mathematical  calculation  it  is  usual  to  still  further  recast 
the  readings  obtained  in  the  vertical  circle  so  as  to  indicate 
the  angle  between  the  plane  of  reference  and  the  plane  con- 
taining the  pole  and  the  normal,  for  the  face.  To  illustrate, 
suppose  there  are  four  faces  belonging  to  the  same  crystal 
set  or  form  having  the  following  vertical  readings:  37°  28', 
142°  32',  217°  28',  322°  32'.  One  projection  point  falls  into 
each  quadrant.  The  plane  carrying  the  normal  for  the 
second  face  makes  with  our  plane  of  reference  an  angle  of 
37°  28'  but  in  the  second  quadrant.  The  plane  carrying  the 
normal  for  the  third  face  lies  in  the  third  quadrant  and 
makes  an  angle  with  the  plane  of  reference  of  37°  28'  and 
similarly  for  the  fourth  face.  This  angle  37°  28'  is  charac- 


26  CRYSTALLOGRAPHY 

teristic  therefore  not  only  of  the  first  but  also  of  the  second, 
third  and  fourth  faces.  It  is  designated  by  the  Greek  letter 
0.  Before  proceeding  to  the  calculation  of  the  symbols  and 
constants  it  is  necessary  to  first  determine  v0  which  may  be 
subtracted  from  the  readings  for  each  face  on  the  vertical 
circle  and  afterwards  to  determine  the  angle  <£  which  locates 
each  point  in  its  particular  quadrant.  Our  problem  there- 
fore is  to  calculate  the  distance  of  a  point  from  the  right  and 
left  line  of  reference  and  from  the  fore  and  aft  line  of  refer- 
ence. These  values  are  functions  of 
0,  p  and  the  height  of  the  plane  of 
projection  above  the  centre.  If  the 
last  value  be  made  unity  then  the 
fore  and  aft  distance  (x)  from  the 
right  and  left  line  is  sin  <J>.  tan  p  while 
the  right  and  left  distance  (y)  is  the 
cos  0.  tan  p  (Figure  13).  The  deriva- 
Fig  13  tion  of  these  formulae  is  indicated  as 

follows : 

To  calculate  the  lengths  of  x  and  y  for  the  face  11  as  shown 
in  the  gnomonic  projection  (Figure  4)  from  D  draw  DB  and 
DC  normals  to  the  lines  of  symmetry  and  DA  to  the  common 
point  of  these  two  lines.  Let  the  height  of  the  plane  of  pro- 
jection  above  the  crystal  centre  be  unity 

Then  AD  =  tan  p 
Z  BAD  =  0 

BD  x 

— —  =  sin  (b  .*.  =  sin  d> 

AD  tan  p 

and  x  =  sin  0  .  tan  p 

AB       CD  y 

Similarly        —  =  —  =  =  cos  0 

AD       AD         tan  p. 

.'.  y  =  tan  p  cos.  <f> 

By  this  method  the  lengths  are  calculated  mathematically. 
The  values  of  x  for  the  various  faces  are  found  to  be  simply 
related  to  one  another.  They  are  easy  multiples  of  some 
common  number.  That  common  number  for  all  the  values 
of  x  is  p0.  For  any  particular  face  the  number  of  paces  (p) 
may  be  obtained  by  dividing  x  by  p0.  Similar  relations  hold 
between  y,  q0  and  q  —  the  right  and  left  pace  number. 


MATHEMATICAL  CHARACTERISTICS 


27 


These  calculations  for  the  various  faces  may  be  made  without 
any  difficulty  for  each  face  but  when  dealing  with  complex 
crystals  presenting  a  large  number  of  faces  it  is  an  ad- 
vantage to  record  the  results  in  some  definite  form  such  as 
the  following: 

TABLE  ILLUSTRATING  AN  ORDERLY  METHOD  OF  RECORDING  CALCULATIONS 


Face 

No. 

Vertical 

, 

p 

(x)  =  sin0tanP 

PO 

p 

Miller's 
symbol. 

Goldschmidt's 
symbol. 

y  =  cos  0  tan  P 

qo 

q 

1 

1 

62°  08' 

62°  08' 

63°  54' 

1.8049 

1.8049 

Ill 

1 

.9539 

.9539 

1 

2 

242°  08' 

62°  08' 

76°  14' 

3.6098 

1.8049 

2 

221 

2 

1.9078 

.9539 

2 

3 

97°  32' 

82°  28' 

61°  13' 

1.8049 

1.8049 

1 

414 

lj 

.2385 

.9540 

1 

4 

180° 

0° 

43°  39' 

zero 

0 

Oil 

01 

.9539 

.9539 

1 

5 

270° 

90° 

42°  04' 

.9024 

1.8048 

i 

102 

10 

zero 

0 

6 

32°  14' 

32°  14' 

73°  32' 

1.8049 

1.8049 

1 

131 

13 

2.8617 

.9539 

3 

There  are  two  problems  either  of  which  may  be  solved  from 
such  calculations  but  both  cannot  be  solved  simultaneously. 
We  find  the  length  of  the  distance  x  or  y  but  as 

x  =  p.  po 

y  =  q-  qo 

it  is  not  possible  to  determine  p  unless  p0  be  already  known 
or  to  determine  q  unless  q0  be  known.  If  on  the  other  hand 
p  and  q  be  known  p0  and  q0  may  be  determined.  If  we 
measure  a  crystal  for  which  the  constants  po  and  qo  have 
been  determined  then  we  may  wish  to  determine  the  values 
p  and  q  and  in  this  way  find  out  the  symbol  for  any  face. 


28 


CRYSTALLOGRAPHY 


On  the  other  hand  we  may  measure  such  a  crystal  in  order 
by  careful  measurements  and  averages  to  check  still  further 
the  values  p0  and  q0.  In  working  with  crystals  for  which 
the  symbols  and  constants  have  not  already  been  fixed  there 
must  be  an  arbitrary  choice  of  certain  faces  to  which  symbols 
are  assigned  and  then  all  the  other  faces  must  receive  symbols 
in  accord  with  these.  The  values  assigned  to  po  and  q0  are 
based  on  the  averages  for  the  values  calculated  from  the  best 
faces. 

7.  Stereographic  Projection.  The  gnomonic  projection  is 
not  commonly  employed  for  purposes  of  illustration  in  works 
of  reference  on  mineralogy  and  crystallography.  Another 
type  of  projection  more  generally  employed  known  as  the 
stereographic  projection  is  obtained  in  the  following  manner. 
Suppose  a  crystal  to  be  placed  at  the  centre  of  a  sphere. 
From  the  central  point  the  normals  for  the  various  faces 
radiate  and  intersect  the  surface  of  the  sphere.  The  eye  of 
the  observer  is  placed  at  a  point  on  the  sphere  in  line  with 
the  standard  vertical  direction  of  the  crystal.  A  plane  passing 
through  the  centre  of  the  crystal  and  at  right  angles  to  the 
standard  vertical  direction  is  the  plane  of  the  stereographic 
projection.  On  this  plane  the  apparent  position  of  the  various 

points  on  the  sphere  surface 
is  indicated.  All  the  projec- 
tion points  on  the  upper  half 
of  the  sphere  are  contained 
within  a  circle,  corresponding 
to  the  equator.  Those  on  the 
half  of  the  sphere  toward  the 
observer  lie  outside  that  circle. 
The  complete  stereographic 
projection  takes  account  of  the 
points  within  and  without  but 
in  general  use  the  points  lying 
outside  the  circle  are  not  indi- 
cated. Faces  belonging  to  the 
zone  which  intersects  in  edges  parallel  to  the  standard  vertical 
direction  are  represented  by  points  on  the  circle  referred  to. 
For  the  rhombic  system  the  face  001  will  appear  in  the  centre  of 
the  circle.  In  this  type  of  projection  the  zone  lines  appear  in 


oooo 


Fig.  14 


MATHEMATICAL  CHARACTERISTICS  29 

three  forms:  (1)  complete  circle;  (2)  arcs  of  circles;  and  (3)  straight 
lines,  where  the  zone  contains  the  pole.  The  magnitude  of 
the  projection  is  proportional  to  the  radius  of  the  sphere. 
Stereographic  projections  may  be  prepared  by  the  same 
method  followed  for  the  gnomonic  projection.  The  chief  differ- 
ence consists  in  the  fact  that  the  distance  of  the  projection 
point  from  the  centre  of  this  projection  is  represented  by 
tan  f  times  the  radius,  while  for  the  gnomonic  projection  the 
distance  is  tan  p  times  the  height  of  the  plane  of  projection 
above  the  crystal  centre.  Figure 
14  is  a  stereographic  projection  of 
topaz  exhibiting  the  projection 
points  fo.r  the  faces  shown  on 
Figure  5.  The  derivation  of  the 
formula  used  is  indicated  by  the 
following  diagram  (Figure  15): 

Let  E  represent  the  centre  of 
the  crystal  placed  at  the  centre  of 
the  sphere,  AC  the  section  through 

the  plane  of  the  gnomonic  projection,  DHF  a  section  of  the 
sphere  DF  the  plane  of  the  stereographic  projection  and  H  the 
point  of  view  for  the  stereographic  projection.  EGK  is  a  face 
normal  making  the  angle  p  with  the  pole  HEB.  The  normal 
intersects  the  sphere  surface  at  G  and  the  plane  of  the  gno- 
monic projection  at  K.  Its  intersection  point  at  G  will  appear 
to  lie  on  the  stereographic  projection  at  the  point  L. 


In  the  A    EGH,  EG  = 

.-.     Z  EGH  =   ZEHG 

But/  BEG  =   Z  EGH  +   Z  EHG 

/.    Z  EHG  =      - 

The  distance  of  the  point  L  from  the  centre  of  the  projection 
is  represented  by  the  distance  EL. 

EL  p  p 

—  —   =  tan  -  /.  EL  =  tan  -.  radius 

EH  2  2   • 

8.  Co-ordinate    Representation    of    Face    Direction.     It  is 

only  in  recent  years  that  the  gnomonic  projection  has  become 


30  CRYSTALLOGRAPHY 

fundamental  in  the  study  of  crystallography.  Previous  to 
this  the  inclination  of  a  crystal  face  was  indicated  by  reference 
to  three  lines  (four  in  the  case  of  the  hexagonal  system) 
intersecting  in  one  point.  Such  a  system  of  imaginary  lines 
is  known  as  the  axial  cross  or  axes  of  reference.  For  the 
cubic,  tetragonal  and  rhombic  systems  these  three  lines  are 
at  right  angles  to  one  another.  In  the  monoclinic  system 
two  of  them  are  at  right  angles  to  one  another,  the  third 
being  at  right  angles  to  one  of  the  first  pair  but  oblique  with 
regard  to  the  other.  In  the  triclinic  system  all  the  angles  are 
oblique.  For  the  cubic,  tetragonal  and  rhombic  systems  the 
axes  are  so  oriented  that  their  directions  are  vertical,  right 
and  left,  and  fore  and  aft.  The  direction  of  a  plane  is  fixed 
when  we  know  the  relative  lengths  measured  from  the  central 
point  at  which  it  intersects  the  three  axes.  In  the  rhombic, 
monoclinic  and  triclinic  systems  the  planes  intersect  the  three 
axes  in  such  a  manner  that  the  lengths  on  the  three  axes 
measured  from  the  centre  to  the  points  of  intersection  are 
always  irrationally  related  to  one  another.  The  three  axes 
are  said  to  be  unequal.  This  corresponds  to  the  irrational 
relationship  which  we  have  already  observed  for  the  rhombic 
system  in  the  case  of  the  values  p0,  qo  and  r0.  In  certain 
systems  it  is  possible  for  a  plane  to  intersect  two  axes  so  that 
the  lengths  of  the  intercepts  are  rationally  related.  Such 
axes  are  equivalent.  In  the  cubic  system  all  three  axes  are 
of  this  character.  For  the  rhombic,  mono- 
clinic  and  triclinic  systems,  the  fore  and 
aft  axis  is  designated  by  the  letter  a,  the 
right  and  left  by  b,  and  the  vertical  axis 
•b  by  c.  Opposite  ends  of  these  axes  are  in- 
dicated by  plus  and  minus  signs  as  may 
be  observed  in  the  diagram  (Figure  16). 
To  the  angles  between  the  various  pairs 
Fig.  16  of  axes  are  assigned  the  letters  a,  /3  and  7. 

In   the   rhombic   these   are   all   90°.      In 

the  triclinic  system  they  are  all  oblique,  while  in  the  mono- 
clinic  only  /3  is  oblique. 

If  several  faces  of  a  crystal  intersecting  the  same  ends  of 
three  axes  be  moved  parallel  with  themselves  so  as  to  pass 
through  the  same  point  on  one  of  the  axes  it  will  then  be 


MATHEMATICAL  CHARACTERISTICS  31 

found  that  the  lengths  cut  off  on  either  of  the  other  axes  by 
the  various  faces  are  simply  and  rationally  related  to  one 
another.  Only  those  faces  are  possible  on  a  crystal  which 
have  such  peculiar  directions  as  to  be  in  accord  with  this 
generalization,  known  as  the  law  of  rationality  of  axial  param- 
eters. This  limitation  as  to  the  possible  intersections  of 
the  axes  of  reference  also  finds  expression  in  a  very  simple 
form  in  the  rationality  of  the  pace  numbers  as  observed  in 
the  gnomonic  projection. 

9.  Derivation  of  Miller's  Index  Symbols.  In  all  crystals 
except  those  belonging  to  the  cubic  system  it  is  necessary 
to  select  arbitrarily  some  unit  crystal  form  or  ground  form 
before  it  is  possible  to  proceed  to  assign  symbols  to  the  other 
faces  that  are  present  or  to  calculate  crystallographic  con- 
stants. In  determining  symbols  in  a  gnomonic  projection 
this  arbitrary  choice  was  made  by  the  selection  of  a  particular 
projection  point  whose  distances  from  the  right  and  left  and 
fore  and  aft  lines  were  regarded  as  the  unit  paces  p0  and  q0. 
Similarly  when  a  complex  crystal  is  studied  with  regard  to 
the  axes  of  reference  a  certain  form  whose  faces  intersect  the 
three  axes  is  arbitrarily  selected  as  the  unit  or  ground  form. 
In  the  rhombic  system  that  form  has  eight  faces  and  is  known 
as  the  rhombic  pyramid.  Any  other  form  whose  faces  inter- 
sect three  axes  in  this  system  is  a  pyramid  but  the  ratios  of 
the  axial  intersections  differ  for  the  different  pyramids.  The 
face  directions  of  crystals  are  constant  though  the  size  of 
the  face  is  exceedingly  variable.  In  indicating  the  ratio  of 
the  lengths  of  the  intercepts  for  the  unit  pyramid  of  the 
rhombic  system  it  is  customary  to  make  the  length  on  the 
b  axis  unity.  For  topaz  the  ratio  is  a:  b:  c:  :  .5285:  1:  .4769. 
The  calculation  of  the  constants  a:b:c  usually  involves  the 
use  of  spherical  trigonometry.  These  axial  ratios  may  be 
derived  from  the  polar  elements  p0:  qo:  r0.  For  the  rhombic 

system   the   following  values   are   sufficient:    a  =  — ;  b  =  1; 

Po 

c  =  q0.  The  transformations  for  the  various  systems  will 
be  indicated  later. 

In  Figure  17  three  pyramidal  faces  are  shown  intersecting 
the  axes  in  various  ratios.  One  of  them  has  such  a  direc- 
tion that  while  intersecting  the  a  axis  at  the  same  point  as 


32  CRYSTALLOGRAPHY 

the  unit  pyramid  it  intersects  the  b  axis  at  twice  and  the  c 
axis  at  three  times  the  distance  characteristic  of  the  unit 
pyramid  or  ground  form.  The  direction  of  this  face  may  be 
indicated  according  to  the  method  of  Weiss  as  follows, 
la:2b:3c.  For  the  third  pyramid  the  corresponding  values 
are  2a:  Ib:  2c.  Planes  which  intersect  only  two  axes  at 
finite  distances  intersect  the  third  at  an  infinite  distance. 
The  selection  of  the  ground  form  determines  the  axial  ratios 
for  a  particular  crystallized  substance.  The  characteristic 
element  in  the  symbols  of  Weiss  is  the  number  placed 
before  the  letters  a,  b  and  c.  These  numbers  are  known  as 
parameters.  They  are  always  rationally  related  to  one 
another.  The  index  symbols  of  Miller  are  the  simplified 
reciprocals  of  these  parameters  written  in  the  same  order 
as  in  the  symbols  of  Weiss.  Thus  for  2a:  lb:3c  the  param- 
eters are  2.1.3  whose  reciprocals  (J-y-i)  when  simplified 
by  multiplying  by  the  least  common  denominator  (6)  gives 
Miller's  symbol  362.  These  new  values  are  known  as  indices 
and  362  is  the  symbol  for  only  one  face.  If  the  face 
symbol  be  enclosed  in  brackets  (362)  it  becomes  a  general 
symbol  for  a  whole  form,  which  in  the  rhombic  system  is  a 
pyramid  with  eight  faces.  By  placing  upon  the  index  num- 
ber the  appropriate  sign,  plus  or  minus,  to  designate  the  end 
of  the  axis  cut  it  is  possible  by  this  system  to  indicate  each 
of  the  eight  faces  belonging  to  the  crystal 
form  in  question.  The  eight  face  symbols 
of_this_form__are  362,  362,  362,  362,  362, 
362,  362  and  362.  We  have  already  shown 


\ 


\    \  how  the    Goldschmidt    symbols   may  be 

\    \  transformed    into  those  of   Miller.     The 

\  reverse  may  be  accomplished  by  dividing 


the  first  two  indices  by  the  third,   thus 
362   in  the  system  of  Miller  becomes  |3 
.  u  according  to  Goldschmidt. 

10.  General   Properties  of  Polyhedra. 

The  smallest  number  of  plane  surfaces  capable  of  enclosing 
space  is  four.  This  is  crystallographically  represented  by  the 
tetrahedron.  On  the  tetrahedron  there  are  four  faces,  four 
corners  and  six  edges.  The  cube  possesses  six  faces,  eight 
corners  and  twelve  edges.  The  relationship  between  the  num- 


MATHEMATICAL  CHARACTERISTICS 


33 


bers  of  edges,  corners  and  faces  exhibited  by  these  two 
polyhedra  is  a  general  characteristic  for  all  polyhedra  and 
may  be  stated  as  follows: 

Faces  +  corners  =  edges  +  2 

When  an  edge  or  a  corner  on  a  polyhedron  is  removed  by  a 
single  plane  the  process  is  known  as  truncation.  Figures 
18  and  19  show  truncation  of  edges  and  corners  as  applied 


Fig.  18 


Fig.  19 


to  the  cube.  Where  the  truncating  plane  makes  equal  angles 
with  the  planes  forming  the  edges  or  corners  the  process  is 
known  as  symmetrical  truncation.  In  all  other  instances  it 
is  unsymmetrical.  Bevelment  is  the  removal  of  an  edge  by 
two  planes.  It  may  be  symmetrical  or  unsymmetrical. 
Three  or  more  planes  removing  a  corner  give  rise  to  acumina- 
tion.  Figures  20  and  21  show  the  symmetrical  bevelment 


s\ 


Fig.  20 


Fig.  21 


and  acumination  as  applied  to  the  edges  and  corners  of  the 
cube.  The  Miller's  symbol  for  a  face  which  symmetrically 
truncates  an  edge  or  a  corner  is  the  algebraic  sum  of  the 
indices  of  the  faces  which  meet  to  form  the  edge  or  corner 
concerned,  thus,  in  Figure  18  the  symbols  of  the  two  cube 


34  CRYSTALLOGRAPHY 

faces  forming  the  edge  truncated  are  (100)  and  (001),  while 
the  symbol  of  the  truncating  face  is  (101).  Similarly  in 
Figure  19  the  symbols  of  the  three  cube  faces  meeting  to 
form  the  corners  truncated  are  (100),  (010)  and  (001)  and 
the  symbol  of  the  truncating  face  —  the  octahedron  —  is 

an). 


CHAPTER   V 
SYMMETRY   OF   CRYSTALS 

Frequent  reference  has  already  been  made  to  the  sym- 
metrical arrangement  of  the  points  of  both  stereographic 
and  gnomonic  projections.  When  examining  crystals  this 
symmetry  shows  itself  in  a  slightly  different  way.  If  a  plane 
be  passed  through  a  cube  centre  and  parallel  to  a  pair  of  cube 
faces  as  shown  in  Figure  22  it  becomes  at  once  apparent 


Fig.  22 


Fig.  23 


that  for  each 'corner  or  edge  shown  on  one  side  of  that  plane 
there  is  a  corresponding  corner  or  edge  on  the  opposite  side. 
If  any  point  on  the  surface  of  the  cube  be  selected,  a  normal 
drawn  from  that  point  to  this  plane  and  produced  an  equal 
distance  beyond  the  plane  it  will  aways  be  found  to  reach 
a  point  similar  in  situation  to  the  one  from  which  it  started. 
A  plane  which  divides  a  polyhedron  into  two  right  and  left 
parts  which  are  similar  to  one  another  according  to  this 
test  is  said  to  be  a  plane  of  symmetry,  and  the  polyhedron 
is  symmetrical  with  regard  to  that  plane.  In  the  cube 
there  are  three  planes  of  symmetry,  each  passing  through 
the  centre  and  each  parallel  to  a  pair  of  cube  faces.  There  , 
are  also  six  other  planes  of  symmetry  in  the  cube,  each 
plane  passing  through  its  central  point  and  through  two  paral- 
lel edges.  (Figure  23.)  A  plane  of  the  class  first  mentioned 
divides  the  cube  into  two  parallelepipeds.  A  plane  of  the 


36 


CRYSTALLOGRAPHY 


second  class  divides  it  into  two  triangular  prisms.  There 
are  therefore  in  the  case  of  the  cube^  three  planes  of  one 
kind  and  six  planes  of  the  other.  Crystals  being  poly- 
hedral forms  frequently  have  their  faces  so  arranged  as  to 
be  symmetrical  about  one  or  more  planes. 

In  most  cases  the  faces  which  belong  to  a  crystal  form 
are  not  situated  at  the  same  distance  from  the  centre  and 
as  a  result  the  crystals  do  not  exhibit  the  ideal  geometrical 
form  and  are  said  to  be  distorted.  Figure  24  shows  this 


Fig.  24 

distortion  with  reference  to  the  cube.  In  all  such  cases 
the  inclinations  of  the  faces  in  the  distorted  forms  are  identi- 
cal with  those  of  the  ideal  form.  This  is  in  accordance  with 
the  law  of  constancy  of  interfacial  angles.  In  considering 
the  presence  or  absence  of  planes  of  symmetry  in  crystals 
we  always  have  in  mind  the  idealized  crystal  which  would 
result  by  giving  equal  central  distance  to  all  those  faces 
belonging  to  the  same  crystal  form.  Figure  25  represents  an 


Fig.  25 


SYMMETRY  OF  CRYSTALS  37 

idealized  crystal  of  the  mineral  quartz  (Si02),  also  crystals 
of  quartz  in  which  distortion  results  from  the  several  faces 
of  the  same  crystal  form  occurring  at  different  distances 
from  the  centre. 

If  a  cube  be  revolved  about  an  axis  passing  through  the 
central  points  of  two  opposite  faces  it  will  be  observed  that 
after  it  has  been  turned  through  an  angle  of  90°  the  edges 
and  corners  of  the  cube  occupy  the  same  positions  as  did 
other  edges  and  corners  before  the  revoluton  was  begun. 
The  cube  when  Devolved  about  a  line  having  this  direction 
will  present  to  the'^xxbserver  the  same  picture  at  intervals 
of  90°.  There  are  kf  £he.cube  three  axes  of  this  type.  If 
rotated  about  an  axis  passing  through  two  diagonally  opposite 
corners  the  cube  will  present  to  the  observer  the  same  pic- 
ture at  intervals  of  120°  —  three  times  during  the  revolution. 
There  are  four  directions  of  this  type  in  the  cube.  Lastly, 
if  a  cube  be  revolved  about  a  line  passing  through  the 
middle  points  of  two  opposite  and  parallel  edges  it  presents 
to  the  observer  the  same  picture  twice  during  the  revolution. 
In  the  cube  there  are  six  axes  of  this  type.  An  axis  about 
which  a  crystal  may  be  revolved  so  as  to  present  to  the 
observer  the  same  picture  two  or  more  times  during  the 
revolution  is  called  an  axis  of  symmetry.  In  the  cube  there 
are  three  different  kinds  of  axes  of  symmetry.  Those  axes 
which  present  the  same  picture  four  times  during  a  revolu- 
tion are  called  tetrad  axes,  similarly  those  presenting  the 
same  picture  two  or  three  times  are  known 
as  diad  and  triad  axes  respectively.  These 
axes  of  symmetry  are  shown  in  Figure 
26.  Crystals  in  their  idealized  form  are 
usually  symmetrical  about  one  or  more 
axes.  The  kinds  of  axes  of  symmetry  as 
exhibited  in  crystals  are  diad,  triad,  tetrad 
and  hexad.  Pentad  symmetry,  which  is  pig>  26 

so  characteristic  of  living  forms,  has  never 

been  observed  in  the  case  of  crystals  and  indeed  is  not  mathe- 
matically possible.  The  axes  of  symmetry  are  sometimes 
designated  as  binary,  trigonal,  tetragonal  and  hexagonal. 

One  other  element  of  symmetry  is  very  frequently  exhibited 
by  crystals,  viz.,  symmetry  about  a  centre  which  implies  the 


X}^ 

i^^^ipt 
i  ff 


38  CRYSTALLOGRAPHY 

occurrence  on  opposite  sides  of  a  crystal  of  parallel  pairs  of 
faces.  The  complete  symmetry  of  the  cube  may  be  stated 
as  follows: 

(a)  Axes    of   symmetry  —  six    diad,    four    triad    and    three 

tetrad. 

(b)  Planes  of  symmetry  —  six  secondary  planes,  three  chief 

planes. 

(c)  Centre  of  symmetry  present. 

Classes  of  polyhedra  involving  only  the  following  elements  of 
symmetry:  planes,  centre  and  diad,  triad,  tetrad  and  hexad 
axes.  If  we  attempt  to  work  out  mathematically  and  classify 
the  possible  types  of  crystal  symmetry  we  find  that  they 
fall  into  thirty-two  distinct  groups  known  as  classes.  Each 
one  of  these  classes  has  a  symmetry  different  from  every 
other.  )There  are  for  example  certain  crystals  which  do  not 
present"  any  element  of  symmetry  —  these  belong  to  the 
hemihedral  subdivision  of  the  triclinic  system.  On  the  other 
hand  the  crystals  belonging  to  the  main  division  of  the  tri- 
clinic system  are  symmetrical  about  the  centre  only.  In 
these  two  classes  the  crystals  are  of  a  very  low  degree  of 
symmetry.  The  crystals  of  the  cubic  system  presenting  the 
symmetry  already  indicated  for  the  cube  are  more  highly 
symmetrical  than  those  of  any  other  group.  When  first 
this  maximum  number  of  possible  crystal  classes  was  announced 
there  were  certain  classes  which  at  that  time  had  no  known 
representative.  At  the  present  time  representatives  are  known 
for  all  but  one  or  two  of  the  thirty-two  classes. 

There  is  an  interesting  correspondence  between  the  com- 
plexity of  chemical  composition  and  the  degree  of  symmetry 
shown  by  crystals.  Most  of  the  substances  devoid  of  sym- 
metry or  possessing  only  a  centre  of  symmetry  have  very 
complex  chemical  formulae  such  as  strontium  bitartrate, 
Sr(C4H4O6H)2.  4  H2O,  or  albite,  NaAlSi308.  In  contrast 
to  this  many  of  the  substances  crystallizing  in  the  cubic 
system  and  exhibiting  a  high  degree  of  symmetry  are  chemi- 
cally extremely  simple — galena,  PbS,  zinc  blende,  ZnS  and  the 
chemical  elements  copper,  gold,  silver  and  carbon  in  the  form 
of  diamond.  While  one  cannot  fail  to  observe  this  corre- 
spondence between  chemical  simplicity  and  high  symmetry 


SYMMETRY  OF  CRYSTALS 


39 


on  the  one  hand  and  chemical  complexity  and  low  symmetry 
on  the  other,  it  is  only  a  general  relationship  for  there  are 
certain  substances  chemically  complex  showing  high  sym- 
metry and  others  chemically  simple  showing  low  symmetry. 
Crystal  Systems.  An  older  classification  of  crystals  and  one 
still  generally  followed  for  didactic  purposes  subdivides  all 
crystals  into  six  systems.  Each  of  these  systems  contains 
several  classes  so  that  within  a  system  there  are  two  or  more 
classes  of  crystals  each  with  its  own  peculiar  degree  of  sym- 
metry. All  those  crystal  classes  which  belong  to  one  system 
are  closely  related  to  one  another:  (a)  their  forms  may  be 
referred  to  the  same  set  of  axes;  (6)  they  resemble  one  another 
very  closely  in  their  physical  properties;  (c)  it  is  usually 
possible  to  derive  the  crystal  forms  of  the  classes  showing 
lower  symmetry  from  forms  of  that  class  presenting  the  high- 
est symmetry  for  the  system  by  supposing  that  such  forms 
of  lower  symmetry  have  been  obtained  by  an  extension  of 
one-half  or  one-quarter  of  the  faces  of  some  crystal  form 
belonging  to  the  class  within  the  system  which  shows  the 
highest  symmetry;  (d)  sometimes  crystal  forms  which  belong 
to  the  highest  class  of  the  system  occur  on  crystals  in  com- 
bination with  forms  which  are  characteristic  of  one  of  the'less 
symmetrical  classes  of  the  system,  e.g.,  a  crystal  belonging 
to  the  cubic  system  may  be  bounded  by  the  six  planes  of  the 
cube  and  the  twelve  planes  of  the  pyritohedron  as  shown  in 
Figure  27.  Now  the  pyritohedron  is  lower  in  symmetry  than 


Fig.  27 


the  cube  but  the  two  of 'them  occur  together  on  the  same 
crystal  as  a  combination.  The  subdivision  of  the  thirty-two 
classes  into  six  systems  is  so  arranged  therefore  that  all 
those  classes  which  are  placed  under  any  one  system  are  very 
closely  related  to  one  another  although  not  identical  in 
symmetry. 


40  CRYSTALLOGRAPHY 

The  following  are  the  names  of  the  six  systems: 

1.  Cubic,  tesseral,  isometric  or  regular, 

2.  Tetragonal  or  quadratic, 

3.  Hexagonal, 

4.  Rhombic,  orthorhombic  or  trimetric, 

5.  Monoclinic,  clinorhombic,  monosymmetric  or  oblique, 

6.  Triclinic,  asymmetric  or  anorthic. 


CHAPTER   VI 
PHYSICAL   PROPERTIES 

The  physical  properties  in  crystals  vary  with  the  direction 
in  the  crystal  but  any  individual  crystal  will  be  found  to 
present  the  same  degree  of  symmetry  from  the  physical  point 
of  view  which  it  presents  when  geometrically  considered. 
As  a  result  of  this  remarkable  correspondence  between  the 
physical  and  geometrical  symmetry  it  is  frequently  possible 
by  a  physical  examination  of  a  fragment  of  a  crystal,  which 
does  not  show  any  natural  crystal  surface,  to  indicate  the 
system  to  which  it  belongs.  In  the  study  of  thin  sections 
of  rocks  under  the  microscope  the  determination  of  the 
system  for  the  various  mineral  fragments  is  based  almost 
entirely  upon  the  optical  properties  of  the  mineral  fragment 
in  question.  In  general  it  may  be  stated  that  directions  which 
are  geometrically  equivalent  in  a  crystal  are  also  physically 
equivalent.  The  cohesion  displayed  by  a  crystal  varies  with 
the  direction.  A  crystal  of  rock  salt  whose  common  form 
is  the  cube  splits  readily  in  three  directions  parallel  to  the. 
faces  of  the  cube.  A  cube  of  fluorite  on  the  other  hand  is 
easily  cleaved  along  four  planes,  each  cleavage  being  parallel 
to  a  plane  which  symmetrically  removes  a  corner  of  the 
cube.  In  the  first  of  these  examples  it  will  be  observed  that 
the  three  directions  of  minimum  cohesion  are  identical  with 
the  directions  of  the  three  tetrad  axes  while  in  fluorite  the 
directions  of  minimum  cohesion  coincide  with  the  four  triad 
axes.  The  particles  of  which  the  crystal  is  composed  appear 
to  be  so  arranged  and  to  possess  such  cohesive  properties  as 
in  themselves  to  be  prophetic  of  both  the  geometrical  form 
and  the  physical  properties  displayed  by  the  crystal. 

If  we  consider  crystals  with  regard  to  their  optical  proper- 
ties we  observe  that  certain  of  them  are  singly  refracting 
without  regard  to  the  direction  in  which  the  light  travels. 


42  CRYSTALLOGRAPHY 

Such  crystals  always  possess  geometrical  forms  which  place 
them  in  the  cubic  system.  All  other  crystals  are  doubly 
refracting  although  there  are  certain  directions  within  them 
in  which  light  may  travel  without  suffering  double  refrac- 
tion. These  directions  of  single  refraction  in  doubly  refract- 
ing substances  are  known  as  optic  axes.  Some  crystals 
possess  only  one  optic  axis  while  others  possess  two.  The 
former  are  geometrically  either  hexagonal  or  tetragonal, 
the  latter  rhombic,  monoclinic  or  triclinic. 

It  is  not  intended  here  to  pursue  the  discussion  of  the 
physical  properties  of  crystals  but  only  to  emphasize  the  fact 
that  the  symmetry  displayed  in  the  geometrical  forms  of 
crystals  corresponds  to  that  revealed  by  their  physical 
investigation. 


CHAPTER  VII 

THEORY  OF  THE  INTERNAL  STRUCTURE  OF 
CRYSTALS 

Certain  marked  regularities  in  crystals  have  already  been 
indicated  and  the  scientific  mind  attempts  to  frame  for  itself 
an  hypothesis  of  a  wider  range  so  as  to  bring  all  these 
observations  into  accord  with  some  fundamental  conception. 
The  regularities  referred  to  may  be  grouped  under  the  follow- 
ing headings: 

1.  Rationality  of  axial  parameters, 

2.  Symmetry  of  crystals, 

3.  Constancy  of  interfacial  angles, 

4.  Variation  of  the  velocity  of  light  within  a  crystal  with 

its  vibration  direction, 

5.  General  purity  of  crystals. 

To  account  for  these  regularities  an  hypothesis  has  been 
formed  which  supposes  that  crystals  are  made  -up  of  small 
particles  or  molecules  which  are  not  necessarily  identical 
with  the  molecules  of  the  chemist.  The  particles  are  regu- 
larly arranged  in  lines  at  equal  distances  from  one  another, 
parallels  of  such  lines  form  planes  while  parallel  planes  of 
such  particles  build  up  the  whole  solid  structure.  These 
crystal  particles  are  held  together  by  mutually  attractive 
forces.  During  the  building  of  the  crystal  there  are  repel- 
lant  forces  which  prevent  the  mingling  together  of  particles 
of  different  kinds  in  the  construction  of  the  same  crystal. 
This  net  work  of  lines  of  particles  is  spoken  of  as  the  lattice. 
For  certain  classes  of  crystals  the  space  distances  between 
the  particles  in  the  three  directions  may  be  identical  as  in 
the  cubic  system.  In  other  systems  these  distances  may  be 
different  from  one  another.  The  angles  formed  by  the 
intersection  of  the  main  lines  of  the  lattice  may  be  rec- 
tangular or  oblique,  thus  giving  rise  to  many  different  lattice 


44  CRYSTALLOGRAPHY 

structures.     According   to   this   hypothesis   the   crystal   faces 
are  planes  of  points. 

In  a  rock  such  as  hornblende  granite  the  arrangement  of 
the  crystal  grains  is  such  that  we  know  that  the  chief  minerals 
-  quartz,  orthoclase  and  hornblende  —  were  all  crystallized 
at  the  same  time.  The  fused  magma  from  which  these  sub- 
stances crystallized  must  have  been  exceedingly  complex. 
The  simultaneous  assembling  of  myriads  of  particles  of  three 
different  kinds,  their  orientation  and  separation  so  as  to  form 
three  different  complex  crystallized  substances  has  a  very 
good  parallelism  drawn  from  military  life.  In  a  garrison  town 
where  are  stationed  infantry,  cavalry  and  artillery  the  men 
at  certain  hours  are  allowed  to  wander  with  the  greatest 
freedom.  At  a  given  signal  they  assemble  the  infantry  in 
one  place  in  regular  order  in  series  of  parallel  lines,  while 
the  cavalry  and  artillery  each  has  its  peculiar  place  of  meet- 
ing and  its  own  order  when  together.  So  do  the  crystal 
particles  of  different  substances  when  taking  on  the  crystal- 
lized condition  gather  about  certain  centres,  orient  themselves 
with  reference  to  the  particles  earlier  fixed,  and  at  the  same 
time  prevent  the  intrusion  of  particles  foreign  to  those  already 
placed. 

All  substances  whose  particles  are  not  arranged  in  a  regular 
manner  are  said  to  be  amorphous.  Substances  of  this  char- 
acter usually  owe  their  form  to  rapid  solidification  caused 
by  sudden  cooling  or  they  may  be  composed  of  several  dif- 
ferent chemical  individuals.  If  quartz  or  garnet  be  fused 
and  the  molten  mineral  suddenly  cooled  by  pouring  it  upon 
an  anvil,  the  quickly  cooled  substance  is  amorphous  and  less 
dense  than  the  crystallized  minerals  from  which  they  are 
formed. 

Crystal  of  quartz  S.G.  2.66  —  when  fused  S.G.  2.20 
"  garnet  S.G.  3.63  -       "         "      S.G.  2.95 
Amorphous  products  obtained  from  the  fusion  of  crystallized 
substances  are  frequently  more  easily  attacked  by  corrosives 
than    the    same    suk/3tances    in    the    crystallized    condition. 
The  particles  are  not  so   closely  packed  together  and  as  a 
result  occupy  a  larger  space  and  are  more  easily  detached. 

The  theory  of  the  internal  crystal  structure  has  developed 
gradually.  It  was  first  suggested  by  Bergmann  under  the 


INTERNAL  STRUCTURE  OF  CRYSTALS 


45 


46  CRYSTALLOGRAPHY 

name  of  the  corpuscular  hypothesis.  Haiiy's  observations  on 
the  cleavage  of  certain  substances  caused  him  to  suggest 
that  crystals  are  built  up  of  particles  of  the  form  enclosed 
by  their  most  prominent  cleavages.  Figure  28  is  from  his 
"Traite  de  Mineralogie,"  1801,  and  indicates  his  application 
of  this  hypothesis  to  the  rhombic  dodecahedron,  pyritohedron 
and  scalenohedron. 

When  crystals  are  grown  together  so  that  only  part  of 
their  faces  are  developed  the  substance  is  said  to  be  crystal- 
lized. In  the  formation  of  rocks  crystallization  frequently 
begins  about  a  multitude  of  centres  and  the  crystallized 
masses  afterwards  come  into  contact  with  one  another  so 
that  when  the  whole  mass  is  solidified  there  are  no  crystal 
faces  to  be  observed.  The  rock  is  granular  and  made  up 
of  grains  in  each  of  which  the  particles  are  all  of  one  kind 
and  regularly  oriented  as  shown  by  an  examination  of  thin 
sections  with  a  microscope.  If  the  mineral  be  cleavable  the 
bright  reflecting  cleavage  planes  are  readily  observed  on 
the  rock  surface.  Such  a  substance  is  said  to  be  crystalline. 
Masses  of  halite  taken  from  some  salt  mines  show  this 
structure.  Most  rock  minerals  are  crystalline;  crystallized 
specimens  showing  partial  enclosure  by  crystal  faces  are  not 
common,  while  the  formation  of  crystals  completely  enclosed 
by  plane  surfaces  is  rather  rare.  In  all  three  types  the 
internal  structure  is  identical. 


PART   II 

CHAPTER   VIII 
CUBIC   SYSTEM 

Within  each  system  there  are  incuded  several  classes  of 
crystals  each  of  which  has  its  own  peculiar  degree  of  sym- 
metry. It  is  usual  when  speaking  of  the  degree  of  symmetry 
of  a  system  to  mention  the  symmetry  which  is  characteristic 
of  that  class  possessing  the  highest  degree  of  symmetry. 
Within  a  system  this  class  is  said  to  be  holohedral,  be- 
cause from  the  forms  belonging  to  it  the  forms  of  certain 
other  classes  may  be  derived  by  selecting  one-half  or  one- 
quarter  of  the  faces.  In  contrast  to  the  holohedral  class 
the  other  classes  belonging  to  the  same  system  are  said  to 
be  hemihedral,  tetartohedral  or  hemimorphic.  The  exact 
meaning  of  these  terms  will  be  made  plain  when  it  is  in  order 
to  describe  the  various  subordinate  classes  belonging  to  the 
systems. 

Symmetry  of  the  Cubic  System.  The  symmetry  of  the 
holohedral  class  of  the  cubic  system  has  already  been  indica- 
cated  in  connection  with  our  study  of  the  symmetry  of  the 
cube.  Crystals  belonging  to  this  class  exhibit  the  following 
elements  of  symmetry: 

(a)  Planes.     Three  chief  planes  of  symmetry,  each  of  which  is  parallel 
to   two   cube   faces,    and  six  secondary   planes  of  symmetry,   each 
passing  through  the  centre,   and  through  two  diagonally  opposite 
parallel  edges  of  the  cube. 

(b)  Axes.    Three  tetrad,  four  triad,  six  diad. 

(c)  Centre  present. 

Standard  Orientation  of  Cubic  Crystals.  Crystallographers 
have  arbitrarily  selected  for  each  type  of  crystal  a  definite 
orientation  which  is  always  followed  for  crystals  of  that 
particular  class.  In  the  cubic  system  the  crystal  is  so  oriented 


48  CRYSTALLOGRAPHY 

that   its   tetrad   axes   are   fore   and   aft,    right   and   left   and 
vertical. 

Axes  of  Reference.  The  axes  of  reference  for  this  system 
consist  of  three  lines  at  right  angles  to  one  another,  one  fore 
and  aft,  one  right  and  left,  the  third  vertical.  In  crystals  of 
all  classes  of  the  cubic  system  the  faces  are  inclined  to  the 
axes  in  such  a  manner  that  the  lengths  of  the  intercepts  on 
all  three  axes  are  simply  and  rationally  related  to  one  another. 
This  is  the  only  system  of  crystals  in  which  the  axes  are 
intersected  by  planes  in  this  peculiar  manner.  Where  all 
three  of  the  axes  of  reference  are  intersected  by  crystal 
planes  at  distances  from  the  centre 
which  are  simply  and  rationally  related 
to  one  another,  it  is  usual  to  speak  of 
the  axes  as  equivalent.  Figure  29 
a  2  shows  the  axial  cross  for  the  cubic 
system. 

All  three  axes  are  designated  by  the 
letter  a,  since  they  are  equivalent  axes, 
Fi    29  but  to  distinguish  them  from  one  an- 

other, the  fore  and  aft  axis  is  desig- 
nated by  ai,  the  right  and  left  by  a2  and  the  vertical  by  a3. 
Classification  of  all  Possible  Planes  Intersecting  Three 
Equivalent  Axes  at  Right  Angles  to  One  Another.  A  plane 
may  have  such  a  direction  with  regard  to  the  axes  of  reference 
that  it  intersects  only  one  axis,  being  parallel  to  the  plane 
marked  out  by  the  other  two.  Two  of  the  three  axes  may  be 
intersected  by  a  plane  at  finite  distances,  while  the  third 
axis  is  intersected  only  at  infinity.  It  is  possible  to  further 
subdivide  this  type  of  intersection  into  two  cases: 

(a)  The  intercepts  of  the  two  axes  are  equal,  and 
(6)  The  intercepts  of  the  two  axes  are  unequal. 
There  are  planes  which  have  such  directions  as  to  intersect 
all  three  axes,  but  here  again  subdivision  is  possible: 
(a)  All  three  intercepts  are  equal, 
(6)   All  three  intercepts  are  unequal, 

(c)  Two  intercepts  are  equal,  the  third  greater,  and 

(d)  Two  intercepts  are  equal,  the  third  less. 

In  considering  therefore  a  classification  of  all  the  possible 
modes  of  intersection  of  planes  on  three  equivalent  axes  at 


CUBIC  SYSTEM  49 

right  angles  to  one  another  we  find  that  there  are  seven 
possibilities.  Remembering  the  degree  of  symmetry  which  is 
characteristic  of  cubic  crystals  it  readily  follows  that  when 
these  axes  are  intersected  by  a  plane  there  are  a  certain 
number  of  other  planes  which  are  necessarily  present  in 
order  that  the  degree  of  symmetry  of  the  system  may  be 
maintained,  e.g.,  if  the  axes  be  intersected  at  equal  distances 
from  the  centre  by  a  plane  it  is  necessary  in  accordance  with 
the  symmetry  that  seven  other  planes  should  also  be  present 
for  this  particular  form.  This  gives  rise  to  the  octahedron, 
as  represented  in  Figure  30.  Where  all  three  axes  are  inter- 


Fig.  30  Fig.  31 

sected  at  different  distances  from  the  centre,  symmetry 
requires  the  presence  of  forty-seven  other  planes.  The 
form  resulting,  known  as  the  hexoctahedron,  is  illustrated  in 
Figure  31.  Corresponding  to  the  possibilities  referred  to 
there  are  seven  crystal  forms  which  may  be  indicated  as 
follows: 

1.  One  axis  cut  —  the  cube, 

2.  Two  axes  cut  equally  —  the  rhombic  dodecahedron, 

3.  Two  axes  intercepted  unequally  —  the  tetrahexahedron 

or  pyramidal  cube, 

4.  Three  axes  cut  equally  —  the  octahedron, 

5.  Three  axes  cut  unequally  —  the  hexoctahedron, 

6.  Two  axes  cut  equally,  the  third  greater  —  the  trisocta- 

hedron, 

7.  Two   axes   cut   equally,   the  third  less  —  the  icositetra- 

hedron. 

These  are  the  seven  crystal  forms  characteristic  of  the  cubic 
system.  Certain  of  them  are  invariable  in  their  form  (1,  2 
and  4)  while  others  are  more  or  less  variable  since  where 
it  is  merely  stated  that  two  or  three  axes  are  intersected  at 


50  CRYSTALLOGRAPHY 

unequal  distances  from  the  centre  there  are  a  multitude  of 
crystallographic  possibilities.  In  the  hexoctahedron  the  planes 
may  have  such  directions  as  to  intersect  the  three  axes  in 
various  ratios  such  as  the  following: 


2a2,  3a3, 
2ai,  3a2.,  5a3, 
4ai,  5a2,  6a3. 

These  are  all  hexoctahedra.  On  diamond  crystals  the  follow- 
ing hexoctahedra  are  found:  (321),  (431),  (541),  (651). 
Such  types  as  the  hexoctahedron  are  said  to  be  variable  forms. 

Cube.  The  cube  is  bounded  by  six  planes  at  right  angles 
to  one  another  meeting  to  form  twelve  edges  and  eight 
trihedral  corners  (Figure  24).  In  the  cube  of  geometry 
these  six  faces  are  all  equidistant  from  the  centre  and  as  a 
result  the  six  planes  are  squares.  In  crystals  on  the  other 
hand  these  planes  frequently  are  at  different  distances  from 
the  centre  and  as  a  result  the  six  surfaces  are  seldom  squares. 
They  are  rectangles  of  varying  proportions.  The  departure 
from  the  idealized  geometrical  form  as  a  result  of  varying 
central  distances  fc/r  planes  belonging  to  the  same  crystal 
form  is  called  distortion. 

The  eight  sectants  into  which  space  is  divided  by  the  three 
chief  planes  of  symmetry,  each  of  which  includes  two  of  the 
axes  of  reference,  are  known  as  octants.  Figure  22  indicates 
this  arrangement  of  the  octants  with  regard  to  the  cubic 
system.  In  the  formation  of  the  symbol  to  designate  a  whole 
crystal  form  it  has  been  agreed  that  the  face  symbol  em- 
ployed for  that  purpose  shall  belong  to  a  face  contained  in 
the  right  front  upper  octant.  Where,  as  in  the  case  of  the 
cube  and  the  rhombic  dodecahedron,  several  faces  appear 
within  the  octant,  the  selection  is  still  further  limited  by  the 
requirement  that  the  face  must  be  one  of  those  intersecting 
the  fore  and  aft  axis  at  the  shortest  distance.  Where  there 
are  two  faces  intersecting  this  axis  at  equal  distances  the 
lower  one  as  they  lie  in  the  octant  is  selected  to  give  the  form 
symbol.  By  this  agreement  all  crystallographers  use  the 
same  face  symbol  for  the  derivation  of  the  form  symbol. 
The  parameter  ratios  for  the  front  face  of  the  cube  are 
lai,  oo  a2,  °o  a3  so  that  the  Miller's  symbol  for  this  face  would 


CUBIC  SYSTEM 


51 


be  represented  by  100  which,  when  enclosed  in  brackets 
thus  (100),  stands  as  the  symbol  for  the  whole  crystal  form. 
The  minerals  pyrite,  FeS2,  fluorite,  CaF2,  and  galena,  PbS, 
frequently  crystallize  as  cubes. 

Rhombic  Dodecahedron.  As  its  name  implies,  this  form 
is  bounded  by  twelve  faces,  each  of  which  is  a  rhombus 
(Figure  32).  These  faces  meet  to  form 
twenty-four  edges,  six  tetrahedral  corners 
marking  the  points  of  exit  of  the  tetrad 
axes  of  symmetry,  and  eight  trihedral  cor- 
ners at  the  point  of  exit  of  the  four  triad 
axes  of  symmetry.  The  angles  over  the 
edges  between  the  faces  are  all  equal  and 
measure  60°  (outer  angles).  The  para- 


Fig.  32 


meter  ratios  for  the  faces  of  the  rhombic  dodecahedron  are  as 

follows: 

lai,  Ia2,  oo  a3 

This  corresponds  to  (110)  in  the  index  symbols  of  Miller. 

Octahedron.  This  is  one  of  the  simplest  forms  of  the  cubic 
system,  inasmuch  as  its  faces  intersect  all  three  axes  at  equal 
distances  from  the  centre.  As  its  name  implies  it  is  composed 
of  eight  faces  intersecting  in  twelve  edges  and  six  tetra- 
hedral corners.  In  the  idealized  form  where  all  the  faces 
have  the  same  central  distance  each  face  is  an  equilateral 
triangle  (Figure  30).  The  parameter  ratios  for  a  face  of  this 

form  are: 

lai,  Ia2,  Ia3, 

so   that   the   Miller's   symbol   for   the   whole   form   is    (111). 

Examples:  Pyrite,  FeS2,  diamond,  alum,  KA1(SO4)2.12H2O. 
Tetrahexahedron  or  Pyramidal  Cube.  This  form  resembles 

in  appearance  a  cube  on  each  of  whose  six  faces  has  been 
placed  a  flat,  four-sided  pyramid  (Figure 
33).  It  is  bounded  by  twenty-four  faces 
which  meet  to  form  twelve  cubic  edges  which 
are  parallel  to  the  edges  of  the  cube  and 
twenty-four  dodecahedral  edges  which  are 
approximately  parellel  to  the  edges  of  the 
rhombic  dodecahedron.  There  are  six 
tetrahedral  corners  marking  the  points 


khO 


Fig.  33 


of  exit  of  the  tetrad  axis  and  eight  hexahedral  corners  indicat- 


52  CRYSTALLOGRAPHY 

ing  the  position  of  the  four  triad  axes.  In  the  idealized  form 
where  all  the  faces  possess  the  same  central  distance,  each  of 
its  twenty-four  faces  is  an  isosceles  triangle.  Each  of  the  faces 
of  this  form  intersects  two  of  the  axes  of  reference  at  finite  dis- 
tances which  are  unequal.  We  may,  therefore,  anticipate  a 
great  variety  of  tetrahexahedra,  all  conforming  with  the  above 
general  description.  The  two  axes  intersected  may  exhibit 
such  ratios  as  the  following: 


2a2,  0°  a3, 
lai,  3a2,  0°  a3, 
2ai,  3a2, 


The  Miller's  symbols  for  these  three  types  are  (210),  (310), 
(320).  On  crystals  of  the  mineral  fluorite  the  following 
tetrahexahedra  have  been  observed:  (610),  (510),  (920), 
(410),  (310),  (520),  (730),  (210)  and  (530). 

General  Symbols.  In  order  to  write  a  symbol  which  will 
stand  for  all  possible  tetrahexahedra,  it  is  necessary  to  intro- 
duce certain  letters,  which  according  to  convention  vary 
within  certain  limits.  It  has  been  agreed  that  the  letters 
h,  k  and  1  shall  represent  indices  with  the  relative  values 
h>k>l.  Where  it  is  only  necessary  to  use  two  of  these  in 
order  to  write  a  symbol,  h  and  k  are  the  ones  selected. 
According  to  this  convention  all  possible  tetrahexahedra  may 
be  represented  by  the  general  symbol  (hkO).  Whenever  we 
wish  to  represent  by  one  general  symbol  all  the  possible 
variations  of  a  variable  form  it  is  necessary  to  use  the  letters 
h,  k  and  1. 

Hexoctahedron.  (Figure  31.)  This  form,  as  its  name  sug- 
gests, is  bounded  by  six  times  eight  faces  —  forty-eight  in  all. 
On  no  other  crystal  form  does  such  a  large  number  of  faces 
occur.  The  number  of  faces  in  the  form  is  due  to  the  high 
symmetry  of  the  cubic  system,  and  to  the  fact  that  its  faces 
are  planes  which  meet  all  three  axes  at  different  intercepts. 
In  the  cubic  system  a  crystal  face  with  such  a  direction  as  to 
intercept  the  three  axes  unequally,  requires  the  simultaneous 
occurrence  of  forty-seven  other  faces,  similarly  oriented  with 
regard  to  the  axes.  In  the  idealized  form  each  of  the  forty- 
eight  faces  is  a  scalene  triangle.  There  are  seventy-two 
edges  —  twenty-four  cubic,  twenty-four  octahedral  and  twenty- 


CUBIC  SYSTEM  53 

four  rhombic  dodecahedral.  These  terms  are  applied  to  the 
various  edges  on  account  of  their  lying  approximately  in  line 
with  the  characteristic  edges  of  the  cube,  the  octahedron  and 
rhombic  dodecahedron.  There  are  corners  of  three  types,  six 
octahedral  corners  marking  the  point  of  exit  of  the  tetrad 
axes,  eight  hexahedral  corners  at  the  exit  of  the  triad  axes 
and  twelve  tetrahedral  corners  indicating  the  point  of  exit 
of  the  diad  axes.  The  hexoctahedron,  like  the  tetrahexahe- 
dron,  is  a  variable  form.  Since  according  to  its  definition 
it  is  only  necessary  that  a  face  of  the  hexoctahedron  intersect 
the  three  axes  unequally  it  is  possible  to  conceive  of  a  large 
number  of  ratios  for  the  lengths  of  the  three  intercepts. 
On  crystals  of  diamond,  as  has  already  been  stated,  the 
following  hexoctahedra  have  been  observed:  (321),  (431), 
(541),  (651).  The  general  symbol  representing  these  and  all 
other  possible  hexoctahedra  is  (hkl). 

Trisoctahedron.  Crystals  of  this  form  resemble  closely 
the  octahedron  and  may  be  regarded  as  an  octahedron  on  each 
of  whose  faces  a  flat,  three-sided  pyramid  has  been  placed, 
as  illustrated  in  Figure  34.  The  trisoctahedron  is  bounded 
by  twenty-four  planes,  which  in  the  idealized 
form  appear  as  isosceles  triangles.  There 
are  two  kinds  of  edges,  the  octahedral  and 
the  rhombic  dodecahedral.  Six  hexahedral 
corners  and  eight  trihedral  corners  mark  the 
points  of  exit  of  the  tetrad  and  triad  axes 
respectively.  The  general  symbol  for  all 
possible  trisoctahedra  is  (hhk);  (221)  has  F-  34 

been    observed    on    diamond    and    cuprite, 
(331)    on    native    silver    and    cuprite,    (661)    on    pyrite. 

Icositetrahedron.  This  form  is  also  known  as  the  leucito- 
hedron  and  trapezohedron.  One  of  these  names  is  given  to  it 
on  account  of  its  being  the  characteristic  form  for  the  mineral 
leucite,  while  the  last  is  suggested  by  the  geometric  outline 
of  the  plane.  There  are  twenty-four  faces,  three  in  each 
octant  —  twenty-four  octahedral  edges  and  twenty-four  cubic 
edges,  three  kinds  of  corners,  six  tetrahedral  marking  the 
positions  of  the  tetrad  axes,  twelve  tetrahedral  at  the  point 
of  exit  of  the  diad  axes  and  eight  trihedral  marking  the 
middle  point  of  each  octant.  Like  the  last  three  forms  the 


54 


CRYSTALLOGRAPHY 


icositetrahedron  is  variable  and  the  letters  h  and  k  must 
be  utilized  in  writing  a  general  symbol  (hkk)  for  all  possible 
icositetrahedra.  The  forms  (811),  (411),  (311)  and  (211) 
occur  on  crystals  of  native  gold.  Other 
minerals  on  whose  crystals  the  icositetra- 
hedron is  frequently  observed  are  garnet, 
native  copper  and  native  silver  (Figure  35). 
It  might  appear  at  first  sight  that  there 
would  be  found  on  crystals  a  very  great 
variety  of  representatives  of  any  one  vari- 
able form.  As  a  matter  of  observation,  how- 


Fig.  35 


ever,  we  learn  that  face  indices  larger  than  ten  are  very  uncom- 
mon in  crystal  symbols.  In  the  trisoctahedron,  for  example,  as 
a  result  of  this  the  actual  number  of  variations  for  (hhk)  is 
not  great  and  on  all  crystals  showing  this  form  measured  up 
to  the  present  time  it  is  probable  that  not  more  than  a  dozen 
distinct  forms  have  been  observed.  This  limitation  is  due 
to  the  rarity  on  crystals  of  such  faces  as  involve  indices 
larger  than  ten. 

Combinations  of  Crystal  Forms.  Crystals  which  are  com- 
pletely enclosed  by  the  faces  of  a  single  crystal  form  are 
very  unusual.  Much  more  frequently  on  the  crystal,  two 
or  more  forms  are  to  be  observed,  the  faces  of  the  one 


Fig.  36  —  Garnet    showing    the  Fig.  37  —  Cuprite    showing   the 

rhombic    dodecahedron     and     the  cube,  a,  (100)  the  rhombic  dodeca- 

icositetrahedron  (211).  hedron,  d,  (110)  and  the  icositetra- 

hedron, 0,  (322). 

form  removing  the  corners  or  edges  of  the  other.  The  com- 
bination of  these  different  forms  is  always  such  that  the  direc- 
tions of  symmetry  of  one  form  are  coincident  with  those 
of  any  other  form  occurring  upon  that  crystal.  Figures  19, 
20,  36  and  37  illustrate  some  of  the  simplest  combinations 
for  the  cubic  system. 


CUBIC  SYSTEM  55 

Limiting  Forms.  Attention  has  already  been  called  to  the 
fact  that  three  of  the  seven  crystal  types  characteristic  of 
the  cubic  system  are  invariable.  It  is  not  necessary  in  writing 
their  symbols  to  make  use  of  the  letters  h,  k  and  1.  In  each 
of  the  other  four  types,  however,  there  is  considerable  varia- 
tion. For  each  of  the  forms  observed  a  definite  symbol 
can  be  written,  but  a  formula  representing  in  a  general  way 
all  possible  tetrahexahedra  involves  the  use  of  the  letters 
h  and  k.  Some  of  the  tetrahexahedra  occur  in  forms  which 
closely  approximate  the  cube.  For  these  the  four-sided 
pyramids  which  rest  on  each  cube  face  are  very  flat.  For 
such  forms  the  intercept  of  one  axis  is  very  large  compared 
with  that  of  the  other,  that  is,  h  is  very  much  greater  than  k. 
Other  tetrahexahedra  seem  to  resemble  more  closely  in  outline 
the  rhombic  dodecahedron.  In  them  the  four-sided  pyramid 
which  rests  on  each  of  the  cube  faces  is  comparatively 
sharp.  This  is  due  to  an  approximate  equality  between  the 
lengths  of  the  intercepts  on  the  two  axes.  Here  it  may  be 
observed  that  the  extremely  flat  tetrahexahedron  approaches 
the  cube  in  form  while  the  steep-faced  tetrahexahedron 
approximates  the  rhombic  dodecahedron.  By  writing  one 
above  the  other  the  symbols  for  these  forms,  thus: 

(100) 
(hkO) 
(HO) 

the  relationship  of  the  three  forms  to  one  another  may  be 
made  plain.  The  ratio  between  h  and  k  is  variable.  In 
certain  forms  h  and  k  are  very  nearly  equal  as  would  be  the 
case  for  the  form  (760),  in  this  way  giving  rise  to  a  tetra- 
hexahedron very  close  to  the  rhombic  dodecahedron.  On  the 
other  hand  where  h  is  very  much  larger  than  k  we  may 
obtain  a  symbol  such  as  (910)  which  approaches  very  near 
to  the  cube  symbol  (100).  We  say,  therefore,  that  the 
tetrahexahedron  is  a  form  varying  between  the  cube  and  the 
rhombic  dodecahedron.  These  two  are  said  to  be  its  limiting 
forms.  Similarly  the  icositetrahedron  approaches  the  octahe- 
dron when  h  and  k  are  nearly  equal  to  one  another  and  the 
cube  when  h  is  much  larger  than  k.  The  limiting  forms  for 
the  trisoctahedron  are  the  rhombic  dodecahedron  and  the 


56  CRYSTALLOGRAPHY 

octahedron.  In  the  hexoctahedron  (hkl)  where  all  three 
indices  differ  in  value  we  have  a  large  number  of  possible 
variations,  thus: 

(a)  When  1  becomes  zero  we  obtain  (hkO)  —  the  tetrahex- 

ahedron, 

(6)  When  1  increases  in  value  until  equal  to  k  we  obtain 

(hkk)  —  the  icositetrahedron, 
(c)  When  k   increases   so   as   to  be 
equal  to  h  we  obtain  the  form 

(hhk)          (hko)  (hhk)  —  the  trisoctahedron. 

/    ^hki)^    \  *n  ^e  study  °f  limiting  forms  of  the 

/  \         cubic  system,  therefore,    we  find  that 

(in) — < — (hkk)— > — doo)    the   hexoctahedron   reaches   its   limits 
Fig.  38  in  three  other  variable  forms,  each  of 

which  varies  between  two  fixed  limits. 
This  is  expressed  diagrammatically  in  Figure  38. 

HEMIHEDRAL  CLASSES  OF  THE  CUBIC  SYSTEM 

The  name  hemihedrism  is  applied  to  certain  forms  within 
a  crystal  system  which  may  be  derived  by  the  selection  of 
one-half  the  number  of  faces  present  on  one  of  the  holo- 
hedral  forms  and  by  their  extension  until  they  intersect 
in  edges  and  corners.  The  forms  which  are  characteristic 
of  three  of  the  subordinate  classes  of  the  cubic  system 
can  be  obtained  in  this  way.  The  fourth  or  tetartohedral 
class  presents  only  such  forms  as  might  be  obtained  by  a 
regular  selection  and  extension  of  one-quarter  of  the  faces 
belonging  to  certain  forms  of  the  holohedral  class.  The 
symmetry  of  the  subordinate  classes  in  any  system  is  always 
lower  than  that  of  the  chief  or  holohedral  class.  The  hemihe- 
dral  and  tetartohedral  forms  usually  occur  on  crystals  in 
combination  with  certain  of  the  forms  which  are  characteristic 
of  the  holohedral  class.  Only  certain  of  the  forms  of  the 
holohedral  class  are  capable  of  giving  rise  to  hemihedral  or 
tetartohedral  forms.  The  cube  for  example  possesses  six 
planes  and  it  is  impossible  to  select  one-quarter  of  these. 
The  hexoctahedron  on  the  other  hand  with  its  forty-eight 
planes  may  be  extended  in  quarters  and  halves  so  as  to  give  rise 
to  several  new  forms,  hemihedral  and  tetartohedral  in  character. 


CUBIC  SYSTEM 


57 


PLAGIOHEDRAL  HEMIHEDRAL  CLASS 

In  the  hexoctahedron  there  are  six  faces  in  each  octant, 
none  of  which  is  common  to  two  octants.  If  we  select 
alternately  one-half  of  the  faces  of  the  hexoctahedron  and 
extend  them  to  enclose  space  we  find  we  obtain  two  new 


Fig.  39 

forms  as  represented  in  Figure  39.  These  new  forms  are 
known  as  pentagonal  icositetrahedra. 

Each  of  these  forms  is  bounded  by  twenty-four  planes 
each  of  which  is  irregularly  pentagonal.  These  two  forms 
are  similar  to  one  another  but  by  no  ingenuity  can  they  be 
so  set  up  as  to  present  exactly  the  same  picture  to  the  ob- 
server. They  are  related  to  one  another  in  the  same  way  as 
the  right  hand  is  to  the  left  or  as  the  mirror  image  is  to  the 
picture  of  the  object  as  viewed  directly.  Such  pairs  of  forms 
are  said  to  be  enantiomorphous.  The  symmetry  displayed 
by  these  plagiohedral  forms  is  considerably  l^ss  than  that 
of  the  holohedral  division  of  the  system  to  which  they  belong. 
A  careful  examination  will  show  that  no  face  is  represented 
by  a  corresponding  parallel  and  opposite  face  —  the  centre 
of  symmetry  has  disappeared.  All  the  planes  of  symmetry 
have  vanished  while  the  axes  of  symmetry  characteristic  of 
the  holohedral  class  all  remain.  The  symmetry  of  the  plagio- 
hedral class  is  therefore  three  tetrad  axes,  four  triad  axes 
and  six  diad  axes.  These  forms  have  been  observed  on  the 
mineral  cuprite,  Cu2O,  where  they  occur  in  combination  with 
the  cube  and  the  octahedron  (Figure  40).  It  is  probable  also 
that  the  minerals  halite,  NaCl,  sylvite,  KC1,  sal-ammoniac, 
NH4C1  and  cerargyrite,  AgCl,  belong  to  this  class. 

The    Miller's    symbols    for    hemihedral    and    tetartohedral 


58  CRYSTALLOGRAPHY 

forms  are  obtained  by  writing  the  symbol  for  a  particular 
face,  enclosing  it  in  brackets  and  placing  in  front  of  the 
bracket  a  Greek  letter  which  is  character- 
istic for  the  particular  class  of  hemihed- 
rism.  For  the  plagiohedral  class  this  let- 
ter is  7  so  that  the  symbols  for  the 
two  pentagonal  icositetrahedra  are  7(hkl) 
(Figure  39a)  and  7(hlk)  (Figure  39b). 
These  are  known  as  the  left  and  right 

pentagonal  icositetrahedra  respectively, 
showing    the    cube,    a,    *          &  .     ^  / 

(100);  the  octahedron,  The  method  employed  in  the  selection 
o,  (111);  and  the  pen-  of  half  the  faces  of  the  hexoctahedron 
tagonal  icositetrahed-  would  not  give  us  forms  geometrically 
ron,  x,  7  (986),  accord-  distinct  from  those  of  the  holohedral 
class  if  applied  to  the  other  six  forms 

of  the  cubic  system.     Let  us  note  that  this  method  of  selec- 
tion involves: 

(a)  The  extension  of  one-half  the  faces  within  the  octant, 
and 

(6)  Suppression  of  the  other  half. 

It  is  therefore  inapplicable  to  the  cube,  octahedron, 
rhombic  dodecahedron,  trisoctahedron  and  icositetrahedron, 
because  for  all  of  these  forms  the  number  of  faces  contained 
within  the  octant  is  odd,  so  that  the  selection  of  one-half  is 
not  possible.  In  the  tetrahexahedron  there  are  contained 
within  the  octant  six  faces  but  it  is  not  possible  to  secure  the 
suppression  of  one-half  the  faces  because  each  face  is  common 
to  two  adjacent  octants.  The  face,  therefore,  which  is  sup- 
pressed in  one  octant  is  reproduced  by  the  extension  of  the 
half  of  it  which  is  selected  for  exten- 
sion in  the  adjacent  octant.  This  is  il- 
lustrated in  Figure  41.  As  previously 
mentioned  the  pentagonal  icositetrahedra 
occur  on  crystals  in  combination  with 
various  forms  characteristic  of  the  holo- 
hedral class  of  the  cubic  system.  The 
internal  molecular  arrangement  which  Fig.  41 

gives  rise  to  the  pentagonal  icositetrahe- 
dra appears  to  be  able  to  express  itself  in  such  forms  as  the 
cube,  the  octahedron  or  the  tetrahexahedron.     These  forms, 


CUBIC  SYSTEM 


59 


however,  are  in  certain  respects  physically  distinct  from  the 
corresponding  forms  in  the  holohedral  division.  We  there- 
fore refer  to  them  as  hemihedral  cubes,  octahedra,  etc. 

PENTAGONAL  HEMIHEDRAL  CLASS 

The  characteristic  forms  belonging  to  this  class  of  hemihe- 
drism  may  be  obtained  by  selecting  alternately  one-half  the 
faces  of  certain  holohedra.  While  the  method  of  selection  of 
the  faces  within  the  octant  is  the  same  as  for  the  plagiohe- 
dral  class  the  difference  lies  in  that  for  the  plagiohedral 
class  the  alternation  was  strictly  continued  when  passing 
from  one  octant  to  the  next,  while  in  the  pentagonal  class 


Fig.  42 

adjacent  faces  in  contiguous  octants  are  either  both  extended 
or  both  suppressed  as  is  shown  in  Figure  42. 

As  a  result  of  this  difference  in  the  selection  the  three 
chief  planes  and  centre  of  symmetry  are  retained  in  the  forms 
belonging  to  the  pentagonal  class  of  hemihedrism.  Only 
certain  holohedra  give  rise  to  new  hemihedral  forms  of  the 
pentagonal  class.  Since  one-half  the  faces  within  the  octant 
are  to  be  selected  only  those  holohedra  which  contain  within 
each  octant  an  even  number  of  faces  need  be  examined. 
Both  the  hexoctahedron  and  the  tetrahexahedron  produce 
hemihedral  forms  by  this  method  of  selection.  The  symmetry 
of  the  pentagonal  forms  is  represented  by  four  triad  axes, 
three  diad  axes,  three  planes  of  symmetry  and  the  centre. 
On  account  of  the  presence  of  the  centre  of  symmetry  the 
faces  of  these  forms  occur  in  parallel  pairs  and  this  class  of 
hemihedrism  is  sometimes  referred  to  as  parallel-faced.  It 
is  also  known  as  pyritohedral  on  account  of  pyrite  being 


60 


CRYSTALLOGRAPHY 


one  of  the  commonest  minerals  to  crystallize  with  this  sym- 
metry. Since  all  three  of  these  names  begin  with  the  letter 
p  the  Greek  letter  TT  is  placed  before  the  Miller's  symbol  for 
the  pentagonal  forms. 

Diploid.  By  the  application  of  the  pentagonal  method  of 
selection  to  the  hexoctahedron  there  result  two  twenty-four 
faced  forms  resembling  somewhat  the  icositetrahedra.  In 
each  octant  there  are  three  four-sided  faces.  The  cubic  and 
octahedral  edges  are  present.  There  are  three  kinds  of 
corners,  eight  trihedral  corners  marking  the  middle  point  of 
the  octants,  six  tetrahedral  corners  of  one  kind  and  twelve 
of  another,  the  former  marking  the  points  of  exit  of  the  diad 
axes,  the  latter  occupying  a  position  in  the  planes  of  sym- 
metry and  approximately  midway  between  the  axes  of  sym- 
metry. If  on  the  hexoctahedron  the  face  hkl  be  marked 
as  one  of  the  twenty-four  to  be  extended,  the  form  represented 
in  Figure  43b  will  be  obtained.  On  the  other  hand  if  hlk 


Fig.  43 

and  the  other  twenty-three  faces  which  must  be  selected  at 
the  same  time  be  extended,  a  slightly  different  form  will  be 
obtained,  as  represented  in  Figure  43a.  These  forms  are  known 
as  diploids.  It  may  be  remembered  that  in  the  plagiohjsdral 
class  a  pair  of  forms  was  obtained  by  starting  from  the 
hexoctahedron.  These  two  were  similar  to  one  another  but 
were  characterized  by  a  right  and  left  handed  arrangement 
so  that  it  is  impossible  by  any  method  of  orientation  to 
bring  the  two  into  a  position  of  absolute  parallelism.  Such 
forms  are  said  to  be  enantiomorphous.  The  two  forms 
obtained  by  the  application  of  pentagonal  hemihedrism  to 
the  hexoctahedron  are  not  so  related  to  one  another.  Geo- 
metrically they  are  not  only  similar  but  identical.  If  one 


CUBIC  SYSTEM  61 

of  these  forms  be  turned  through  an  angle  of  90°  about  any 
one  of  its  diad  axes  it  will  then  be  in  all  respects  parallel 
to  the  other  form.  Here  the  two  forms  differ  from  one 
another  only  in  orientation  and  are  said  to  be  congruent. 
They  are  distinguished  as  positive  and  negative.  The  former 
has  the  symbol  Tr(hkl),  the  latter  Tr(hlk). 

Pyritohedron.  As  a  result  of  the  application  of  this  method 
of  selecting  one-half  the  faces  to  the  tetrahexahedron  two 
new  forms  are  obtained  known  as  pyritohedra  or  pentagonal 
dodecahedra  (Figure  42);  the  former  name  has  reference  to 
the  frequent  occurrence  of  the  mineral  pyrite  in  these  forms, 
while  the  latter  becomes  at  once  apparent  by  reference  to 
the  number  and  outline  of  the  faces.  The  pyritohedron  is 
bounded  by  twelve  planes,  each  approximately  pentagonal 
in  outline.  There  are  thirty  edges,  twenty-four  of  which 
radiate  in  sets  of  three  from  the  middle  of  the  octant  while 
the  other  six  occupy  positions  corresponding  to  the  middles 
of  the  cube  faces.  The  corners  are  all  trihedral.  Eight  of 
them  mark  the  central  point  of  the  octant  and  twelve  have 
positions  approximately  indicated  by  the  point  of  exit  of  the 
diad  axes  for  the  holohedral  class.  The  two  forms,  designated 
as  positive  and  negative,  bear  the  same  relation  to  one  another 
as  the  corresponding  diploids.  Symbols: 

1.  Positive—  Tr(hkO) 

2.  Negative -Tr(khO) 

The  diploids  and  pentagonal  dodecahedra  are  commonly 
found  on  the  minerals  of  the  pyrite  and  alum  groups.  The 
following  are  a  few  of  the  commoner  diploids  observed  on 
pyrite:  x(321),  7r(432),  ir(751),  ir(531)  and  7r(532).  On  the 
same  mineral  the  pyritohedra  7r(410),  ?r(310),  7r(520)  and 
7r(210)  are  frequent.  There  is  an  interesting  geometrical 
form  bounded  by  twelve  regular  pentagons.  In  many  re- 
spects it  is  geometrically  closely  related  to  certain  pyritohe- 
dra, but  no  pyritohedron  exactly  of  this  form  has  ever  been 
observed.  If  such  a  form  were  referred  to  three  equivalent 
axes  at  right  angles  to  one  another  its  indices  would  be 

2.  (  -          - ).  0,  which  would  be  in  conflict  with  the  law  of  the 

\          a          / 

rationality    of    axial    parameters,    which    permits    only    such 


62 


CRYSTALLOGRAPHY 


Fig.  44 


faces  on  crystals  as  give  indices  simply  and  rationally  related 
to  one  another. 

The  application  of  this  method  of  selection  to  the  other 
five  holohedral  forms  would  not  produce  types  geometrically 
,  different.  The  diploids  and  pyritohedra 
occur  in  combination  with  one  another 
and  also  in  combination  with  these  five 
forms.  When  this  occurs  we  regard  the 
latter  as  being  physically  hemihedral 
although  geometrically  identical  with  the 
corresponding  holohedral  forms.  The 
mineral  pyrite  frequently  occurs  in  cubes 
which  are  striated  in  such  a  manner  that 
the  striae  are  parallel  to  the  cube  edges 

and  the  sets  of  striations  on  adjacent  faces  are  at  right  angles 
to  one  another  (Figure  44).  This  immediately  informs  us  that 
the  lines  representing  the  diagonals  of  the  cube  face  are  not  lines 
of  symmetry  as  they  are  in  the  case  of  an  ordinary  cube. 
A  cube  so  striated  is  symmetrical  about  only  three  planes, 
each  of  which  is  parallel  to  a  pair  of  cube  faces.  If  a  crystal 
be  immersed  in  some  solvent  or  corrosive  and  withdrawn 
after  a  very  small  amount  of  its  material  has  been  removed, 
it  will  usually  be  found  that  its  faces  are  marked  by  a  number 
of  corrosion  pits.  These  etching  figures  or  corrosion  pits 
indicate  a  certain  degree  of  symmetry  and  on  any  one  crystal 
face  their  corresponding  directions  are 
parallel.  A  cube  of  sylvite  corroded 
in  this  way  gives  rise  to  four-sided 
pits  arranged  as  indicated  in  Figure  45. 
While  these  little  pits  are  symmetri- 
cal about  certain  planes  these  planes  do 
not  coincide  in  direction  with  the  geo- 
metrical planes  of  symmetry  character- 
istic of  the  cube.  In  this  way  we  are 
able  to  determine  that  the  cube  of  syl- 
vite is  not  physically  in  conformity  with  the  symmetry  of  the 
holohedral  division  of  the  cubic  system.  This  method  of  deter- 
mining the  physical  symmetry  of  crystals  will  be  considered  in 
detail  later.  It  is  consequently  to  be  expected  that  crystals 
which  exhibit  the  characteristic  hemihedral  forms  of  a  particular 


Fig.  45 


CUBIC  SYSTEM  63 

class  will  also  exhibit  those  forms  of  the  holohedral  class  which 
do  not  give  geometrically  new  types  by  the  application  of 
this  method  of  selection.  Figures  46  and  47  exhibit  com- 


Fig.  46  —  Pyrite 
showing  the  octahed- 
ron, o,  (111)  and  the 
pyritohedron,  e, 
ir(210). 


Fig.  47  — Pyrite  ex- 
hibiting the  octahed- 
ron and  pyritohedron 
in  combination  in  such 
a  development  as  to 
simulate  the  geomet- 
rical icosihedron. 


Fig.  48  —  Pyrite  show- 
ing the  combination  of  the 
cube,  a,  (100)  and  the 
diploid,  s,  7r(321). 


binations  of  the  octahedron  and  pyritohedron.  In  the  latter 
combination  there  is  a  close  approximation  to  the  geometrical 
icosihedron  which  is  bounded  by  twenty  equilateral  triangles. 
Figure  48  represents  the  cube  in  combination  with  the 
diploid.  Figure  2  shows  a  very  common  combination  of  the 
cube  and  pyritohedron. 


TETRAHEDRAL  HEMIHEDRAL  CLASS 

The  crystal  forms  characteristic  of  this  class  may  be  simply 
derived  by  extending  the  faces  of  the  holohedral  forms  con- 
tained in  four  alternate  octants.  All  those  holohedra  whose 
faces  are  confined  to  a  single  octant  give  new  hemihedral 
forms  by  this  method.  The  holohedral  forms  whose  faces 
occur  in  only  a  single  octant  are  the  following:  octahedron, 
hexoctahedron,  trisoctahedron  and  icositetrahedron.  The 
forms  belonging  to  this  class  are  characterized  by  six  planes 
of  symmetry,  three  diad  axes  and  four  triad  axes.  As  they 
are  not  symmetrical  about  a  centre  the  faces  do  not  occur 
in  parallel  pairs.  The  letter  K  is  the  characteristic  for  this  class. 

Tetrahedron.  In  the  octahedron  only  one  face  is  con- 
tained within  the  octant.  The  extension  of  one-half  of  the 


64 


CRYSTALLOGRAPHY 


octants,  i.e.,  one-half  of  the  faces  of  the  octahedron  is  illus- 
trated   in    Figure    49.     There    are    therefore    two    congruent 


Fig.  49 

tetrahedra,  one  of  which  results  by  the  extension  of  the  faces 
which  are  suppressed  in  the  first  instance.  These  forms  are 
positive  and  negative  with  the  following  symbols: 

1.  Positive  —  K (111) 

2.  Negative  — K  (ill) 

The  tetrahedron,  as  its  name  implies,  is  bounded  by  four 
planes,  each  face  in  the  idealized  form  being  an  equilateral 
triangle.  There  are  four  trihedral  corners  and  six  edges. 

Trigonal  Tristetrahedron.  Two  trigonal  tristetrahedra  may 
be  obtained  by  the  application  of  this  method  of  selection  to 
the  icositetrahedron.  In  outline  they  resemble  a  tetrahedron 
on  each  of  whose  faces  a  flat,  three-sided  pyramid  has  been 
placed  in  the  manner  indicated  in  Figure  50.  The  trigonal 


Fig.  50 

tristetrahedron  is  enclosed  by  twelve  faces,  each  of  which 
is  an  isosceles  triangle.  As  the  icositetrahedron  is  a  variable 
form  so  is  the  trigonal  tristetrahedron.  The  forms  are  posi- 
tive and  negative  and  possess  the  following  symbols: 


CUBIC  SYSTEM 

1.  Positive--   K(hkk) 

2.  Negative  —  K  (hkk) 


65 


Tetragonal  Tristetrahedron.     The  same  method  applied  to 
the  trisoctahedron  gives  rise  to  the  forms  shown  in  Figure  51. 


Fig.  51 

They  resemble  the  tetrahedron  in  outline,  but  the  flat,  three- 
sided  pyramid  is  turned  in  a  different  direction  from  that 
characteristic  of  the  last  form.  Each  of  the  twelve  faces  is 
deltoid  resembling  in  outline  the  faces  of  the  icositetrahedron. 
These  congruent  forms  are  positive  and  negative  with  the 
following  symbols: 

1.  Positive —   /c(hhk) 

2.  Negative  —  x(hhk) 

Hextetrahedron.  By  applying  the  tetrahedral  method  of 
selection  to  the  hexoctahedron  two  new  forms  result  which 
bear  to  one  another  the  relationship  which  characterized 
the  three  pairs  of  the  previous  forms  (Figure  52).  In  out- 


Fig.  52 

line  the  hextetrahedron  suggests  a  tetrahedron  on  each  of 
whose  faces  has  been  placed  a  flat,  six-sided  pyramid.  Each 
of  the  twenty-four  faces  is  a  scalene  triangle.  There  are  three 
kinds  of  corners,  four  hexahedral  marking  the  central  point 
of  the  tetrahedral  face,  four  hexahedral  at  the  points  corre- 


66 


CRYSTALLOGRAPHY 


spending  to  the  corners  of  the  tetrahedron  and  six  tetrahedral 
corners.  They  are  variable  forms  and  being  positive  and 
negative  have  the  following  symbols: 

1.  Positive--    /c(hkl) 

2.  Negative  —  K  (hkl) 

The  various  new  forms  resulting  from  the  tetrahedral 
method  of  selecting  one-half  the  faces  of  the  holohedra  occur 
in  congruent  pairs  which  are  geometrically  identical  and 
differ  from  one  another  only  in  orientation.  These  forms 
occur  in  combination  with  one  another  and  with  those 
holohedra  of  the  cubic  system  which  do  not  give  rise  to 
geometrically  new  types  as  a  result  of  this  method  of  hemihe- 
dral  selection.  Combinations  of  hemihedral  and  holohedral 
types  are  shown  in  Figure  53  indicating  simple  combination 


Fig.  53 


Fig.  54  — The  cube,  a,  (100); 
the  rhombic  dodecahedron,  d,  (110) 
and  the  tetrahedron,  o,  jc(lll). 


of  the  cube  and  tetrahedron,  and  in  Figure  54  exhibiting  the 
combination  of  the  forms  on  the  last  figure  with  the  addi- 
tion of  the  rhombic  dodecahedron.  Sphalerite,  tetrahedrite 
and  diamond  are  among  the  best  known  minerals  crystalliz- 
ing in  this  class. 

TETARTOHEDRAL  CLASS 

There  are  certain  crystal  forms  which  contain  one-quarter 
of  the  faces  of  some  one  of  the  holohedral  forms  of  the  cubic 
system  and  which  may  be  regarded  as  derived  by  the  simul- 
taneous application  of  any  two  of  the  above  described  hemihe- 
dral methods. 

It  might  be  thought  at  first  glance  that  a  great  variety  of 
tetartohedral  forms  would  result  by  this  method,  but  a  care- 


CUBIC  SYSTEM 


67 


ful  examination  indicates  that  it  is  only  the  hexoctahedron 
which  gives  new  forms  as  a  result  of  the  simultaneous  applica- 
tion of  two  hemihedral  methods.  Indeed,  geometrically  the 
same  forms  are  obtained  regardless  of  which  pair  of  hemihe- 
dral methods  is  applied.  The  surviving  faces  of  the  hexocta- 
hedron and  the  method  of  selection  are  indicated  in  Figures 
55  and  56.  Three  faces  survive  in  alternate  octants  so  that 


Fig.  55 

the  new  tetartohedral  form  resembles  in  a  way  the  tetrahedron 
each  of  whose  faces  is  covered  by  a  low,  three-sided  pyramid. 
The  new  form  is  bounded  by  twelve  planes,  each  irregularly 
five-sided  and  is  known  as  the  tetrahedral  pentagonal  dodeca- 
hedron. As  indicated  in  the  above  figures  two  enantio- 


Fig.  56 

morphous  forms  are  obtained  when  three  of  the  surviving 
faces  belong  to  the  front,  right  upper  octant.  There  is  also 
a  pair  of  enantiomorphous  forms  resulting  when  three  of  the 
faces  to  be  extended  occur  in  the  left,  front  upper  octant. 
Theoretically  therefore  there  are  four  tetrahedral  pentagonal 
dodecahedra.  Each  form  of  the  pair  first  mentioned  is  con- 
gruent with  a  form  of  the  second  pair.  While  therefore  we 
have  crystallographically  four  forms,  geometrically  there  are 
only  two.  They  are  designated  as  follows: 


68 


CRYSTALLOGRAPHY 


1.  Positive    right-handed  —  K7(khl) 

2.  Negative  right-handed  —  K7(hkl) 

3.  Positive    left-handed      —  K7(hkl) 

4.  Negative  left-handed      —  K7(khl) 

Forms  1  and  2  are  congruent.     The  same  relationship  holds 
between  3  and  4. 


Fig.  57 — Barium  nitrate  showing 
the  cube,  a,  (100);    the  tetrahedron, 

0,  K  (ill) ;  the  trigonal  tristetrahedra, 

1,  K  (311)  and  s,  K  (2ll);  and  the  tet- 
rahedral   pentagonal    dodecahedra, 
t,  Ky  (214),  h,  Ky  (2l4)  an    n,  Ky  (351). 


Fig.  58  —  Sodium  chlorate  show- 
ing the  cube,  a±  (100)  ;  the  tet- 
rahedron, o,  /c(lll);  the  rhombic 
dodecahedron,  d,  (110)  and  the  pen- 
tagonal dodecahedron,  p,  7r(210). 


The  tetartohedral   class  in   the   cubic   system   is   exhibited 
by  only  a  small  number  of  crystallized  substances,   such  as 


Fig.  59 


CUBIC   SYSTEM 


69 


sodium  chlorate,  sodium  bromate,  and  the  nitrates  of  lead, 
barium  and  strontium.  Figure  57  presents  a  very  complex 
crystal  of  barium  nitrate,  exhibiting  tetartohedral  faces. 
Crystals  on  which  tetartohedral  faces  do  not  occur  may  be 
shown  to  belong  to  this  class  of  symmetry  by  the  simultane- 
ous occurrence  on  them  of  hemihedral  forms  belonging  to 
different  classes,  e.g.,  the  simultaneous  occurrence  on  a  crystal 
of  tetrahedra  and  pyritohedra.  Figure  58  shows  tetartohe- 
drism  of  this  class  for  a  crystal  of  sodium  chlorate. 

Projections.  Figures  59  and  60  show  the  character  of  the 
gnomonic  and  stereographic  projections  respectively  for  the 
crystal  of  galena  represented  in  Figure  61.  From  the  gno- 


Fig.  60  Fig.    61  —  Galena    showing   the 

cube,  a,  (100) ;  the  octahedron,  o, 
(111) ;  the  rhombic  dodecahedron, 
d,  (110)  and  the  icositetrahedron, 
/»,  (322). 

monic  projection  symbols  for  the  various  faces  may  be  readily 
determined.  As  has  already  been  pointed  out  for  the  cubic 
system  the  three  axes  ai,  a2,  a3  being  equivalent  the  polar 
elements  p0,  Qo,  and  r0  are  all  identical.  In  connection  with 
the  cubic  system,  therefore,  the  principal  problem  is  the 
determination  of  the  symbols  for  the  faces  present  upon 
the  crystal. 


CHAPTER   IX 
TETRAGONAL   SYSTEM 

Within  the  tetragonal  system  are  contained  seven  classes 
of  crystals,  each  differing  from  the  others  in  symmetry. 
These  various  classes  are  closely  related  to  one  another  in  the 
following  ways : 

1.  They   are   usually   characterized   by   the   presence   of   a 
single  tetrad  axis  of  symmetry. 

2.  The  faces  occurring  on   crystals  of   the   several    classes 
can  always  be  referred  to  an  axial  cross,  consisting  of  three 
axes  at  right  angles  to  one  another,  two  of  them  equivalent, 
the  third  different. 

3.  As  in  the  cubic  system  there  are  here  certain  holohedral 
forms  which  by  hemihedral  and  tetartohedral  selection  do  not 
give  rise  to  new   geometrical  forms.     The  forms   peculiar  to 
and  characteristic  of  each  class  frequently  occur  in  combina- 
tion with  these  holohedra. 

When  the  tetragonal  system  is  mentioned  the  reference  is 
to  the  holohedral  class  except  where  special  care  is  taken  to 
indicate  one  of  the  classes  of  a  lower  degree  of  symmetry. 

HOLOHEDRAL  CLASS 

Symmetry.  Crystals  belonging  to  this  class  show  the  following 
symmetry : 

1.  Axes.  One  tetrad;  two  diad  at  right  angles  to  one  another 

and  to  the  tetrad  axis;  two  diad  axes  lying  in  the  same 
plane  as  the  previous  pair  of  axes  and  bisecting  the 
angles  between  them. 

2.  Planes.  One  plane  of  symmetry  known  as  the  chief  plane 

at  right  angles  to  the  tetrad  axis,  and  four  planes  of 
symmetry  each  containing  the  tetrad  axis  and  one  of  the 
four  diad  axes  above  mentioned. 

3.  Centre  present. 


TETRAGONAL  SYSTEM  71 

Standard  Orientation.  It  is  customary  to  orient  crystals 
of  the  tetragonal  system  in  such  a  manner  that  the  tetrad 
axis  of  symmetry  is  vertical  and  so  that  two  diad  axes  are 
respectively  fore  and  aft  and  right  and  left.  Since  in  this 
system  there  are  four  diad  axes  lying  in  the  horizontal  plane 
there  are  two  possible  methods  of  orientation  for  each  tetrago- 
nal crystal.  When,  however,  a  crystal  of  the  tetragonal 
system  is  first  measured,  the  crystallographer  selects  one 
pair  of  diad  axes  rather  than  the  other  to  have  the  fore  and 
aft  and  right  and  left  positions,  and  henceforth  the  orientation 
of  crystals  of  that  particular  substance  is  fixed. 

Axes  of  Reference.  In  the  tetragonal  system  there  are 
three  axes  of  reference,  one  vertical,  which  coincides  with 
the  tetrad  axis  and  a  pair  of  equivalent  axes,  one  fore  and 
aft,  the  other  right  and  left,  coinciding  in  direction  with  two 
diad  axes  of  symmetry.  These  two  lateral  axes  are  spoken 
of  as  secondary  axes  of  symmetry  while  the  two  others  lying 
in  the  same  plane  and  midway  between  them  are  known  as 
intermediate  axes  of  symmetry.  The  planes  of  symmetry 
containing  the  vertical  and  lateral  axes  of  reference  are 
known  as  secondary  planes,  while  the  other  two  planes  of 
symmetry  are  referred  to  as  intermediate  planes.  Any 
crystal  face  which  intersects  all  three  axes  of  reference  gives 
intercepts  on  the  lateral  axes  simply  and  rationally  related 
to  one  another.  Either  these  two  axes  are  intercepted  at  the 
same  distance  from  the  centre  or  the  one  is  cut  at  a  finite 
distance,  the  other  at  infinity  or  lastly, 
the  lengths  of  the  intercepts  of  the  two 
axes  are  related  to  one  another  in  some 
simple  ratio  such  as  2:1,  2:3,  or  3:4. 
As  a  result  these  two  axes  are  said  to  be 
equivalent  and  are  designated  ai,  a2,  as 
was  the  case  in  the  cubic  system.  In 
the  tetragonal  system  the  length  of  the 
intercept  on  the  vertical  axis  is  always 
irrationally  related  to  the  length  of  the 
intercept  on  a  lateral  axis.  The  vertical  Fig.  62 

axis  is  therefore  not  equivalent  to  the 

horizontal  and  in  the  axial  cross  is  designated  by  the  letter  c. 
These  relationships  are  exhibited  in  Figure  62. 


72  CRYSTALLOGRAPHY 

Classification  of  All  Planes  with  Reference  to  Their  Inter- 
section of  the  Tetragonal  Axial  Cross.  We  can  simply  divide 
all  possible  planes  on  tetragonal  crystals  into  two  classes. 
Some  have  such  a  direction  that  they  do  not  intersect  the 
vertical  axis  because  there  lies  in  the  plane  a  line  which  is 
parallel  to  this  axis.  All  other  planes  intersect  the  vertical 
axis.  Planes  of  the  former  class  may  be  subdivided  as 
follows: 

(a)  The  plane  intersects  both  lateral  axes 

(1)  Equally, 

(2)  Unequally. 

(6)  The  plane  intersects  only  one  lateral  axis. 
Planes  of  the  second  class  may  be  subdivided  thus: 
(a)  One  lateral  axis  cut  at  a  finite  distance, 
(6)  Two  lateral  axes  cut  at  finite  distances 

(1)  Equally, 

(2)  Unequally. 

(c)   Both  lateral  axes  are  intersected  at  infinity. 
With  regard,  therefore,  to  the  way  in  which  planes  inter- 
sect the  tetragonal  axial  cross  there  are  seven  types,  giving 
rise   to   distinct   crystal   forms.     The   seven   types,    with   the 
names  for  the  tetragonal  system  are  as  follows: 

1.  Vertical  axis  cut  —  lateral  axes  cut  equally  —  pyramid 

of  the  first  order, 

2.  Vertical   axis  cut  —  lateral    axes  cut  unequally  —  ditet- 

ragonal  pyramid, 

3.  Vertical  axis  cut  —  only  one  lateral  axis  intersected  — 

pyramid  of  the  second  order, 

4.  Vertical  axis  cut  —  lateral  axes  not  cut  —  basal  pinacoid, 

5.  Vertical  axis  not  cut  —  lateral  axes  cut  equally  —  prism 

of  the  first  order, 

6.  Vertical    axis    not    cut  —  lateral    axes    cut    unequally  — 

ditetragonal  prism, 

7.  Vertical  axis  not  cut  —  only  one  lateral  axis  cut  —  prism 

of  the  second  order. 

Pyramid  of  the  First  Order.  This  form,  as  may  be  ob- 
served from  the  above  sketch,  is  bounded  by  planes  which 
intersect  the  two  lateral  axes  with  equal  intercepts  and  the 
vertical  axis  with  an  intercept  which  is  irrationally  related 
to  the  intercept  on  the  lateral  axes.  The  symmetry  of  this 


TETRAGONAL  SYSTEM  73 

system  requires  that  when  one  such  plane  appears  seven 
others  appear  on  the  crystal  at  the  same  time.  (Figure  63.) 

Such  a  set  of  faces  is  known  as  a  crystal  form.  There 
are  according  to  the  definition  many  ways  of  intersecting  the 
lateral  axes  equally  and  the  vertical  axis  at 
a  distance  irrationally  related  to  the  lateral 
distances.  The  one  who  first  describes  the 
crystal  of  a  tetragonal  substance  arbitrarily 
selects  some  pyramid  of  the  first  order,  usu- 
ally one  represented  by  large  faces  upon  the 
crystal  and  designates  it  as  the  unit  pyra- 
mid of  the  first  order.  When  this  has  been 
done  the  ratio  between  the  lengths  of  the 
intercepts  on  the  vertical  and  lateral  axes  Fig.  63 

for  a  plane  of  the  unit  pyramid  becomes  a 
characteristic  of  the  substance  in  question.  The  length  of 
the  vertical  intercept  for  this  unit  form  is  indicated  by  the 
letter  c,  that  of  the  lateral  axes  by  the  letter  a.  The  ratio 
for  these  two  unit  lengths  is  expressed  as  follows  for  the 
tetragonal  mineral  zircon: 

c:a  :  :  .640373:  1 

The  vertical  intercept  is  expressed  in  terms  of  the  lateral 
intercept,  a  being  by  agreement  equal  to  1.  Sometimes  this 
is  expressed  still  more  briefly,  thus: 

c  =  .640373 

There  are  certain  pyramids  of  the  first  order  whose  planes 
are  steeper  and  have  an  intercept  on  the  vertical  axis  greater 
than  unity,  while  others  are  flatter  and  possess  intercepts 
on  the  lateral  axes  greater  than  unity.  The  symbol  for  the 
unit  pyramid  or  ground  form  is  (111).  The  steeper  pyramids 
may  be  symbolized  as  (hhk)  and  the  flat  pyramids  as  (kkh). 

The  pyramid  of  the  first  order  is  bounded  by  eight  planes 
each  of  which  is  an  isosceles  triangle.  There  are  four  basal 
edges  and  eight  polar  edges.  The  corners  are  all  tetrahedral, 
two  of  them  polar  and  four  basal.  In  a  general  way  these 
pyramids  resemble  the  octahedron,  except  that  the  faces  are 
always  isosceles  triangles  and  not  equilateral  as  in  the  case 
of  the  octahedron. 

Ditetragonal  Pyramid.  This  form  results  from  the  inter- 
section of  the  two  lateral  axes  unequally  and  the  vertical 


74 


CRYSTALLOGRAPHY 


Fig.  64 


axis  at  a  distance  irrationally  related  to  the  lengths  of  the 
lateral  axes.  The  symmetry  of  the  tetragonal  system  re- 
quires that  such  planes  should  appear  in  sets  of  sixteen.  The 
ditetragonal  pyramid  is  bounded  by  sixteen  planes,  each  of 
which  is  a  scalene  triangle.  (Figure  64.)  There  are  eight 
basal  edges  and  two  sets  of  polar  edges  —  eight  in  each  set. 
The  corners  are  of  three  kinds,  two  octahed- 
ral corners,  marking  the  highest  and  lowest 
points  of  the  crystal,  four  tetrahedral  cor- 
ners at  the  points  of  exit  of  the  lateral  axes 
of  reference  and  four  tetrahedral  corners  at 
the  points  of  exit  of  the  intermediate  axes 
of  symmetry.  There  are  many  different  di- 
tetragonal pyramids,  some  flatter  and  some 
steeper,  varying  with  the  relative  length  of 
the  intercept  on  the  vertical  axis.  The  ratio 
between  the  lengths  of  the  intercepts  on  the 
lateral  axes  may  also  vary,  such  values  as  2:  1,  3:  2,  4:  3  being 
among  the  commonest.  It  is  not  difficult  to  determine  the 
symbol  for  any  particular  face  since  the  plane  may  be  sup- 
posed to  be  moved  parallel  to  itself  so  as  to  intersect  one  of 
the  lateral  axes  at  the  unit  distance,  and  the  other  lateral 
at  greater  than  unit  distance,  such  as  two,  three  or  four 
times.  When  this  occurs  the  length  of  the  intercept  on  the 
vertical  axis  will  be  found  to  be  simply  related  to  the  unit 
length  on  that  axis.  Thus  the  parameter  ratios  on  the 
three  axes  may  be  as  follows: 

lai:  2a2:  lc,  or  lai:  3a2:  2c. 

From  these  parameter  ratios  the  Miller's  symbols  may  be 
obtained  as  follows: 

From  lai:  2a2:  lc  the  reciprocals  1:^:1  are  derived. 
When  these  are  simplified  we  obtain  the  Miller's  symbol  (212). 
The  general  symbol  for  all  ditetragonal  pyramids  is  (hkl). 

Pyramid  of  the  Second  Order.  The  faces  of  this  form 
intersect  the  vertical  and  one  lateral  axis,  being  parallel  to 
the  second  lateral  axis.  As  a  result  of  the  symmetry  of  the 
system  the  crystal  form  is  composed  of  eight  faces,  each  of 
which  is  an  isosceles  triangle.  Geometrically  these  pyramids 
are  similar  to  the  pyramids  of  the  first  order.  They  are 
different  however  in  orientation.  (Figure  65.) 


TETRAGONAL  SYSTEM 


75 


According    to  the  standard  orientation  for  the  tetragonal 
system,   the   pyramid   of    the   second   order   presents   a    face 


Fig.  65 


Fig.  66 


toward  the  observer,  while  its  basal  edges  are  right  and  left 
and  fore  and  aft.  The  relative  orientation  of  the  three  te- 
tragonal pyramids  is  indicated  in  Figure  66,  while  the  corn- 


Fig.  67 


Fig.  68 


bination  of  two  orders  of  pyramids  on  cassiterite  is  illustrated 
in  Figures  67  and  68.  The  general  symbol  for  all  pyramids 
of  the  second  order  is  (hOl)  —  for  the  unit  pyramid  (101). 

Basal  Pinacoid.  The  faces  of  the  basal 
pinacoid  are  planes  which  intersect  the  S^  001,°  s^ 
vertical  axis  at  a  finite  distance  but  are 
parallel  to  the  plane  containing  the  lateral 
axes.  As  this  system  is  characterized  by 
the  presence  of  a  centre  of  symmetry  there 
are  two  such  planes  comprised  in  the  crys- 
tal form.  They  do  not  meet  to  form  edges 
and  consequently  do  not  enclose  space. 
Such  a  crystal  form  is  known  as  an  open 
form  in  contrast  to  those  already  studied  Fig.  69 


-a-2 -- 


76 


CRYSTALLOGRAPHY 


which  are  known  as  closed  forms.  (Figure  69.)  The  faces  of  the 
cube,  the  octahedron  or  of  the  tetragonal  pyramids  meet  in 
edges  and  corners  and  enclose  space.  Four  is  the  smallest  num- 
ber of  planes  which  can  enclose  space, 
as  illustrated  by  the  tetrahedron.  Open 
forms  never  occur  on  crystals  alone  but 
must  necessarily  appear  in  combina- 
tion with  other  forms.  Figure  70  in- 


Fig.  70 


-4--1- 


Fig.  71 


dicates  the  combination  of  the  basal  pinacoid  with  the  pyra- 
mid of  the  first  order.     Symbol  (001). 

Prism  of  the  First  Order.  Planes  of  this  form  intersect 
the  lateral  axes  at  equal  distances,  the  vertical  axis  at  in- 
finity. The  symmetry  of  the  system  demands  the  presence 
of  four  such  faces  which  meet  in  four  edges 
parallel  to  the  vertical  axis.  The  prism  of 
the  first  order  is  an  open  form,  square  in 
cross-section.  It,  like  the  basal  pinacoid, 
only  occurs  on  crystals  in  combination  with 
other  forms.  Symbol  (110).  Figure  71 
shows  this  prism  in  combination  with  the 
basal  pinacoid. 

Prism  of  the  Second  Order.  Here  only 
one  lateral  axis  is  intersected  by  the  plane 
and  the  whole  form  is  made  up  of  four 
planes  meeting  in  edges  parallel  to  the  vertical  axis.  The 
angles  between  adjoining  planes  are  90°.  This  prism  is 
related  to  the  prism  of  the  first  order  in  the  same  way  as 
the  pyramid  of  the  second  order  is  related  to  the  pyramid 
of  the  first  order.  Its  symbol  is  (100).  Figure  72  represents 
the  prism  of  the  second  order  combined  with  the 
pyramid  of  the  first  order. 

Ditetragonal  Prism.  Each  face  of  this  form 
intersects  the  two  lateral  axes  at  different  dis- 
tances and  in  simple  ratio,  such  as  2:1,  3:2  or 
4:3.  The  form  is  enclosed  by  eight  planes  in- 
tersecting in  edges  parallel  to  the  vertical  axis. 
Its  cross-section  is  eight-sided  as  shown  in  Fig- 
ure 66.  A  ditetragonal  prism  whose  cross-sec- 
tion would  be  a  regular  octagon  would  involve  irrationality 
of  the  axial  parameters,  and  consequently  does  not  occur 


Fig.  72 


TETRAGONAL  SYSTEM 


77 


T 

.^ 

'- 

—  r 

J_- 

— 

—  L 

—  _^ 

Fig.  73 


on  tetragonal  crystals.  The  ditetragonal  prism  is  a  vari- 
able form.  There  are  numerous  possibilities,  since  it  is 
only  necessary  that  the  ratios  of  the  lateral  axes  should  be 
simply  related  to  one  another.  Its  general 
symbol  is  (hkO).  Figure  73  exhibits  this 
prism  in  combination  with  the  basal  pina- 
coid.  On  the  tetragonal  mineral  rutile  the 
following  ditetragonal  prisms  have  been  ob- 
served: (430),  (320),  (530),  (210),  (940), 
(310),  (410),  (710)  and  (810). 

Calculation  of  the  Ratio  c :  a.  This  may 
be  most  simply  derived  from  the  inner  basal 
angle  between  the  faces  (101)  and  (101)  of 
the  pyramid  of  the  second  order.  The  dia- 
gram (Figure  74)  illustrates  the  intersection  of  such  a  pyra- 
mid by  a  plane  passing  through  the  fore  and  aft  and  verti- 
cal axes.  From  this  diagram  it  is  readily 
apparent  that  the  tangent  of  one-half  basal 
angle  equals  c,  a  being  unity. 

Z  ABC  =  Basal    angle   of    pyramid    of 
second  order, 
AE  and   EB   the  unit   intercepts  on 

the  c  and  a  axes  respectively, 
Z  ABE  =  \  basal  angle, 

Fig.  74 -Section  of  ~  =  ^  =  tan-  z  ABE  =  tan.  J  basal  angle, 

the  unit  pyramid  of  the  1   u        i          i 

second  order  (101)  pass-  "    C  :=  tan'  *  basal  anSle' 

ing  through  the  vertical      This  ratio  may  be  determined  from  vari- 

axis  CA  and  the  axis  ous  other  interfacial  angles.     In  most  other 

ai,    BD.      From    this  cases   it   is  necessary  to  employ  spherical 

diagram  it  is  apparent  trigonometry,  with  which  some  readers  are 

that  c:  a::  AE:  BE:  and  .,,  ,    ,       .,. 

that  c  =  tan  Z  ABE.     P°^ibly  not  familiar : 

The   ratio   c:a   is    a   constant   which   is 

characteristic  for  each  tetragonal  substance.  It  is  identical 
with  the  pace  length  p0  obtained  from  the  gnomonic  projec- 
tion. For  the  tetragonal  system  the  constants  are  thus 
related : 

Po  =  c. 


The  holohedral  class  of  the  tetragonal  system  is  represented 


78 


CRYSTALLOGRAPHY 


by   the   minerals   rutile,    vesuvianite   and    zircon,    which    are 
shown  in  Figures  75,  76  and  77. 


1 


Fig.  75  — Rutile 
showing  the  prism 
of  the  first  order, 
m,  (110);  the  prism 
of  the  second  order, 
a,  (100);  the  pyra- 
mid of  the  first 
order,  s,  (111)  and 
the  pyramid  of  the 
second  order,  e, 
(101). 


Fig.  77 —  Zircon  showing 
the  prism  of  the  first  order, 
m,  (110)  and  the  pyramid  of 
the  first  order,  p,  (111). 


Fig     76  —  Vesuvia- 
nite showing  the  prism 

of  the  first  order,  m, 

(110) ;  the  prism  of  the 

second  order,  a;  (100); 

the  ditetragonal  prism, 

/,  (210) ;  the  basal  pina- 

coid,  c,  (001)  and  the 

pyramids   of   the  first 

order,  b,  (221)  and,  p, 

(111). 

Limiting  Forms  in  the  Tetragonal  System.  There  are  in 
this  system  pyramids  of  three  types,  the  fir^st  order,  the 
second  order  and  the  ditetragonal  pyramids.  Of  each  of 
these  types  there  are  numerous  forms,  some  flatter,  others 
steeper.  As  a  pyramid  becomes  steeper  the  two  faces  which 
meet  to  form  a  basal  edge  approach  more  nearly  to  a  posi- 
tion of  parallelism.  We  may  therefore  regard  the  prism 
as  an  infinitely  steep  pyramid  of  the  same  order.  On  the 
other  hand  the  flattening  of  the  pyramid  causes  the  various 
faces  which  intersect  the  same  end  of  the  vertical  axis  to 
approach  the  parallel  position.  When  this  occurs  the  basal 
pinacoid  is  the  resulting  form.  The  pyramid,  therefore,  of 
whatever  class,  may  be  regarded  as  variable,  swinging  be- 
tween its  limits  —  a  prism  and  the  basal  pinacoid.  So  far 
we  have  merely  considered  the  variations  of  the  intercept  on 
the  vertical  axis.  There  are  two  types  belonging  to  the 
tetragonal  system  which  have  variable  intercepts  on  the 
lateral  axes  —  the  ditetragonal  pyramid  and  the  ditetragonal 
prism.  The  general  symbol  for  the  ditetragonal  prism  is 
(hkO).  Where  k  approaches  h  in  value  we  obtain  a  ditetrago- 


TETRAGONAL  SYSTEM  79 

nal  prism  which  closely  simulates  the  prism  of  the  first  order 
(110).  Where  k  becomes  so  small  as  to  approach  0  a  type 
of  ditetragonal  prism  results  approaching  closely  the  prism 
of  the  second  order.  This  may  be  illustrated  by  the  following 
series  of  symbols  for  the  various  prisms :  (001) 

(110)  —  prism  of  the  first  order, 

(870) 


(210) 
(910) 


—  ditetragonal  prisms, 


<—  (hkl)— -> (hOl) 

(100)  —  prism  of  the  second  order.         f  I  f 

The  same  relationship  maintains  be-   (no)    -<    (hko)  — * — (100) 

tween  the  ditetragonal  pyramid  and  the  Fig.  78 

other  pyramids.    The  limitations  for  the 

various  forms  of  the  tetragonal  system  are  diagrammatically 

shown  in  Figure  78. 

TETRAGONAL  CLASSES  OF  LOWER  SYMMETRY 

Belonging  to  the  tetragonal  system  there  are  in  all  seven 
classes.  The  one  already  described,  commonly  known  as  the 
holohedral  class,  presents  the  highest  symmetry.  From 
forms  of  this  class,  by  selecting  one-half  or  one-quarter  of 
the  faces  in  a  regular  way,  new  forms  of  lower  symmetry  are 
obtained.  These  are  the  characteristic  forms  for  the  various 
classes  of  lower  symmetry.  In  this  respect "  the  tetragonal 
and  cubic  systems  are  analogous. 

TRAPEZOHEDRAL  HEMIHEDRAL  CLASS 

This  class  of  hemihedrism  resembles  in  many  respects  the 
plagiohedral  of  the  cubic  system.  Its  forms  are  derived  by 
the  selection  and  extension  of  one-half  the  faces  within  the 
octant  and  the  simultaneous  suppression  of  the  other  half. 
Forms  geometrically  new  can  only  result  under  these  restric- 
tions when  this  method  is  applied  to  holohedra  whose  faces 
do  not  extend  from  one  octant  into  the  next  and  where  the 
number  of  faces  within  the  octant  is  even.  There  is  only 
one  tetragonal  holohedron  which  fulfils  these  conditions, 
viz.,  the  ditetragonal  pyramid.  Figure  79  illustrates  the 
selection  of  one-half  the  faces  of  this  pyramid  according  to 
the  trapezohedral  method.  Two  enantiomorphous  forms 


80 


CRYSTALLOGRAPHY 


result,  one  when  the  face  hkl  and  the  other  seven  which 
belong  to  this  set  are  extended,  and  a  second  when  the  face 
khl  and  its  set  are  extended.  These  forms,  known  as  tetrago- 
nal trapezohedra,  are  related  in  the  same  way  as  are  the  two 
pentagonal  icositetrahedra  previously  described.  They  are 
devoid  of  centre  and  planes  of  symmetry  but  retain  the  same 
axes  of  symmetry  as  were  characteristic  for  the  holohedral 
class  —  one  tetrad  axis,  and  two  pairs  of  diad  axes  lying  in  a 
plane  at  right  angles  to  the  tetrad  axis.  These  forms  are 
distinguished  from  one  another  as  right  and  left.  The  letter 


Fig.  79 

T  placed    before  the   Miller's   symbol  indicates   this   class   of 
hemihedrism.     The  symbols  for  these  two  forms  are  therefore: 

1.  Left,  r(hkl) 

2.  Right,  r(khl) 

These  peculiar  trapezohedral  forms  have  been  observed  on 
nickel  sulphate,  NiSO4  +  6H20,  and  on  crystals  of  certain 
complex  organic  substances,  such  as  sulphate  of  strychnine. 
They  occur  in  combination  with  the  other  six  forms  of  the 
tetragonal  system  which  do  not  give  rise  to  new  forms  by 
this  method  of  selection.  The  internal  molecular  arrange- 
ment which  finds  expression  in  the  trapezohedral  forms  is 
also  capable  of  building  up  forms  which  are  geometrically 
identical  with  certain  of  the  tetragonal  holohedra.  When 
this  occurs  there  is  always  a  physical  difference  between  the 
holohedral  prism,  pyramid  or  pinacoid,  and  the  geometrically 
identical  trapezohedral  prism,  pyramid  or  pinacoid.  This 
physical  difference  may  be  revealed  by  means  of  etching 
figures,  or  still  more  simply,  by  an  optical  examination.  If 


TETRAGONAL  SYSTEM 


81 


a  thin  plate  be  cut  parallel  to  the  basal  pinacoid  and  examined 
in  polarized  light  it  will  be  found  that  such  plates  cut  from 
trapezohedral  crystals,  even  though  they  do  not  show  the 
characteristic  trapezohedral  forms,  turn  the  plane  of  polari- 
zation to  the  right  or  to  the  left.  In  this  way  we  can  dis- 
tinguish not  only  the  trapezohedral  crystals  from  other 
tetragonal  crystals,  but  we  may  distinguish  right  and  left 
handed  crystals.  There  is,  therefore,  a  physical  difference 
between  prisms  and  pyramids,  which  are  geometrically 
identical. 

PYRAMIDAL  HEMIHEDRAL  CLASS 

This  type  of  hemihedrism  corresponds  in  a  measure  to  the 
pentagonal  hemihedrism  of  the  cubic  system.  One-half  of 
the  faces  within  an  octant  are  selected  but  when  passing 
from  an  upper  to  a  lower  octant  the  alternation  is  not  con- 
tinued. The  alternate  selection  is  strictly  adhered  to  except 
for  this  departure.  The  holohedra  which  give  rise  to  new 
forms  by  this  method  of  selection  must  contain  an  even 
number  of  faces  within  the  octant  and  the  faces  may  not 


Fig.  80 

extend  from  one  upper  octant  into  another  upper  octant  or 
from  one  lower  octant  into  another  lower  octant.  A  face 
may,  however,  extend  from  an  upper  into  a  lower.  With 
these  restrictions,  it  is  readily  apparent  that  the  only  two 
forms  which  can  be  productive  of  geometrically  new  types  are 
the  ditetragonal  pyramid  and  the  ditetragonal  prism.  Figure 
80  indicates  this  method  of  selection  as  applied  to  the  ditetrag- 
onal pyramid.  It  will  be  observed  that  four  faces  survive  from 
the  upper  octants  and  that  the  four  faces  immediately  beneath 


82 


CRYSTALLOGRAPHY 


100- 


represent   the   lower   octants.     Two    congruent   forms   result. 
Geometrically  they  resemble  the  pyramids  of  the  first  and 
j  second  orders  and  are  called  pyramids 

of  the  third  order.  These  three  types 
of  pyramid  differ  from  one  another 
chiefly  in  orientation  as  shown  in  Fig- 
ure 81,  which  represents  diagrammati- 
cally  a  cross-section  of  these  three 
pyramidal  types.  If  one  of  the  pair 
of  congruent  forms  shown  in  Figure 
80  be  turned  about  its  tetrad  axis 
through  a  certain  angle  it  assumes  an 

orientation  identical  with  the  other  form.  They  are  dis- 
tinguished from  one  another  as  right  and  left  and  possess 
the  following  symbols: 

1.  Left  — Tr(hkl) 

2.  Right  —  TT  (khl) 

This  method  of  selection  applied  to  the  ditetragonal  prism 
as  indicated  in  Figure  82  results  in  the  suppression  of  one- 


khO 


hkO 


Fig.  82 


half  the  prismatic  faces  and  the  extension  of  the  other  half. 
Two  new  forms  —  prisms  of  the  third  order  —  are  obtained, 
geometrically  identical  with  the  prisms  of  the  first  and  second 
order  which  differ  from  them  only  in  orientation.  They  are: 

1.  Left  —  Tr(hkO) 

2.  Right  —  Tr(khO) 

The  symmetry  of  forms  belonging  to  this  class  is  as  follows: 


TETRAGONAL  SYSTEM 


83 


One  chief  plane, 
One  tetrad  axis, 
Centre  present. 

The  pyramids  and  prisms  of  the  third  order  occur  in  com- 
bination with  certain  holohedra  of  this  system.  In  this 
as  in  the  last  class  the  physical  properties  of  the  crystals  are 
distinct  from  those  of  the  holohedral  class,  even  though  they 
should  be  bounded  only  by  apparently  holohedral  forms. 
The  minerals  scheelite,  CaWO4,  scapolite  (Figure  83),  fergu- 


Fig.  83  —  Scapolite  showing 
the  prism  of  the  first  order,  m, 
(110);  the  prism  of  the  second 
order,  a,  (100);  the  pyramid 
of  the  first  order,  o,  (111)  and 
the  pyramid  of  the  third  order, 

Z,    7T  (311). 


Fig.  84  —  Ferguso-       Fig.    85  —  Stolzite 
nite  showing  the  prism  showing    the  pyramid 
of  the  third  order,  g,  of  the   first   order,    o, 
TT  (320);   the  pyramid  (111)    and    the    prism 
of  the  third  order,  z,  of  the  third  order,  g, 
TT  (321);  the  pyramid  TT  (430). 
of   the  first  order,  s, 
(111)    and    the   basal 
pinacoid,  c,  (001). 


sonite  (Figure  84)  and  stolzite,  PbWO4  (Figure  85)  crystallize 
in  this  class. 

SPHENOIDAL  HEMIHEDRAL  CLASS 

A  third  hemihedral  class  results  from  the  selection  of  one- 
half  the  faces  by  octants.  Half  the  octants  are  selected 
alternately  and  all  the  faces  contained  within  the  selected 
octants  are  extended.  Those  contained  in  the  other  four 
octants  are  suppressed.  Geometrically  new  forms  may  be 
obtained  here  only  from  those  holohedra  where  faces  are 
confined  to  a  single  octant.  Where  a  face  is  common  to  two 
octants,  new  forms  will  not  result  by  the  application  of  this 
method.  Only  two  holohedra,  therefore,  are  capable  of  giv- 


84  CRYSTALLOGRAPHY 

ing  rise  to  hemihedral  forms  of  this  class,  viz.,  the  pyramid  of 
the  first  order  and  the  ditetragonal  pyramid.     Figure  86  indi- 


Fig.  86 

cates    the    application    of    this    method    of    selection    to    the 
ditetragonal  pyramid. 

From  the  pyramid  of  the  first  order  are  obtained  two  con- 
gruent forms  each  enclosed  by  four  isosceles  triangles.  These 
forms  differ  from  one  another  only  in  orientation,  while  they 
differ  from  the  tetrahedron  in  that  their  faces  are  isosceles  in- 
stead of  equilateral  triangles.  These  hemihedral  forms  which 
are  called  tetragonal  sphenoids  of  the  first  order  are  symbolized 
thus: 

1.  Positive   --/c(hhl) 

2.  Negative  —  /c(hhl) 

Figure  87  represents  a  combination  of  the;  positive  and  nega- 
tive forms. 

From  the  ditetragonal  pyramid  a  pair  of  congruent  forms 
is  obtained.  Each  form  is  bounded  by  eight  faces  and  has  a 


Fig.  87  —  Chalcopyrite  showing 
the  positive  sphenoid  o,  K  (111)  and 
the  negative  sphenoid,  —  o,  K  (111). 


Fig.  88 — Urea  showing  the  prism 
of  the  first  order,  m,  (110);  the 
basal  pinacoid,  c,  (001)  and  the  posi- 
tive sphenoid,  o,  K  (111). 


TETRAGONAL  SYSTEM  85 

general  resemblance  in  its  outline  to  the  tetragonal  sphenoid, 
a  pair  of  faces  here  replacing  a  single  face  of  the  last  type. 
They  are  known  as  tetragonal  scalenohedra  and  are  distin- 
guished from  one  another  as  follows: 

1.  Positive   — jc(hkl) 

2.  Negative  —  *(hkl) 

The  symmetry  for  forms  of  this  class  is: 
Two  intermediate  planes, 
Two  diad  axes  corresponding  in  direction  to  the  axes 

of  reference  ai  and  a2, 

One  diad  axis  parallel  to  the  vertical  axis  c. 
This   type   of    hemihedrism   is   exhibited  on  crystals  of  chal- 
copyrite,   CuFeS2  (Figure  87),  and  of  urea,  CH4N2O  (Figure 
88). 

TETARTOHEDRAL  CLASS 

The  simultaneous  application  of  two  methods  of  hemihe- 
drism gives  rise  to  forms  possessing  one-quarter  of  the  faces 
characteristic  of  certain  holohedra.  Figures  79,  80  and  86 
indicate  the  method  of  selection  characteristic  of  the  three 
hemihedral  processes.  The  simultaneous  application  of  the 
sphenoidal  and  pyramidal  methods  of  selection  to  the  dite- 
tragonal  pyramid  produces  the  bisphenoids  of  the  third  order. 
These  forms  resemble  the  tetragonal  sphenoids  already  de- 
scribed. They  are  four  in  number  according  to  orientation 
and  possess  the  following  symbols: 

1.  Positive  right   -  —  o-(khl), 

2.  Positive  left        -  o-(hkl), 

3.  Negative  right  —  KTr(hkl), 

4.  Negative  left    —  KTr(khl). 

Similarly  from  the  pyramid  of  the  second  order  two  sphe- 
noids of  the  second  order  are  obtained.  This  same  method 
of  selection,  when  applied  to  the  ditetragonal  prism  and  to 
the  pyramid  of  the  first  order,  produces  a  prism  of  the  third 
order  and  the  sphenoid  of  the  first  order  identical  in  all 
respects  with  those  already  described.  Forms  belonging  to 
this  class  are  symmetrical  about  only  one  diad  axis  corre- 
sponding in  direction  with  the  vertical  axis  for  tetragonal 
crystals.  Up  to  the  present  time  no  substances,  either 


CRYSTALLOGRAPHY 


natural   or   artificial,    have   been   recognized   as   belonging  to 
this  class. 

HOLOHEDRAL    HEMIMORPHIC    CLASS 

Certain  substances  crystallizing  in  the  tetragonal  system 
are  peculiar  in  that  the  faces  which  intersect  the  vertical 
axis  at  the  positive  end  do  not  correspond  to  those  inter- 
secting this  axis  at  the  negative  end.  They  are  said  to  be 
hemimorphic  and  the  vertical  axis  is  called  a  hemimorphic 
axis.  There  are  two  classes  of  such  crystals  belonging  to 
this  system.  One,  known  as  the  holohedral  hemimorphic 
e'/ass,  presents  the  following  symmetry: 
Axes.  One  tetrad, 

Planes.     Two  secondary  and  two  intermediate, 
Centre  absent. 

This  class  represents  a  symmetry  which 
would  result  by  starting  with  the  holohed- 
ral class  and  dropping  the  chief  plane  of 
symmetry,  which  would  incidentally  carry 
with  it  the  centre  and  the  four  diad  axes. 
The  name  given  suggests  its  derivation  in 
this  fashion  from  the  holohedral  class. 

The  pyramids  and    pinacoid  of  the  holo- 
hedral  class  fall  into   pairs   of  hemimorphic 
forms,  one  designated  as  upper  and  the  other 
These    may   be   symbolized    as   fol- 


Fig.   89  —  lodo- 
succinirmde    show-    lower- 
ing  the  prism  of  the    lows : 
first  order,  m,  (1 10) ;        Pyramid 
the  upper  and  lower 
halves  of  the  pyra- 
mid   of    the    first 
order,  6,  (221)  and 
(221;;     the     lower 
half  of  the  pyramid 
of  the  first  order,  o, 
(111). 


1.  First  order,  upper  (hhl)  and 

lower  (hhl) 

2.  Second   order,    upper    (hOl) 

and  lower  (hOl) 

3.  Ditetragonal,     upper     (hkl) 

and  lower  (hkl) 

Basal  pinacoids.     The  basal  pinacoid,  up- 
per (001)  and  lower  (OOl). 
lodosuccinimide  crystallizes  in  this  class  (Figure  89). 


HEMIHEDRAL  HEMIMORPHIC  CLASS 

A  second  hemimorphic  class  of  the  tetragonal  system  stands 
in  the  same  relationship  to  the  pyramidal  hemihedral  class 


TETRAGONAL  SYSTEM  87 

as  the  holohedral  hemimorphic  occupies  with  reference  to 
the  holohedral  class.  The  various  forms  other  than  the 
prisms  are  designated  as  upper  and  lower  which  is  shown  in 
the  symbols  by  the  sign  placed  over  the  last  index  to  indicate 
which  end  of  the  vertical  axis  is  intersected  by  the  form  in 
question.  The  tetrad  axis  is  the  only  element  of  symmetry 
characteristic  of  these  forms.  Figure  90 
indicates  the  characteristic  combination 
on  a  crystal  of  wulfenite,  PbMoO4. 

Crystals  of  these  two  hemimorphic 
classes  behave  in  a  peculiar  way  when 
heated  or  cooled.  It  is  found  that  with 
the  change  of  temperature  the  faces 
grouped  around  opposite  ends  of  the  ver- 
tical axis  become  electrified.  In  a  par-  t  FiS-  90  ~  Wulfenite 
..  ,  ,1,1  i  1-11  showing  the  upper  and 

ticular  crystal  that  end  which  becomes  iower  halves  of  the  pyra- 
positively  electrified  with  the  rise  of  tern-  mid  of  the  first  order, 
perature  becomes  negatively  electrified  o,  (111)  and  o'  (111); 
with  the  fall  of  temperature.  The  re-  the  lower  half  of  the 
verse  holds  for  the  other  end.  Such  crys-  Pyamid  of  _the  second 
.  ,  ,  i  j.  •  rrii.  order,  e',  (101);  the  half 

tab  are  said  to  be  pyro-electric.     This    pyramids  rf   the  third 

phenomenon,  which  may  be  observed  in  order,  a?,ir(311)  and  z, 
crystals  of  many  classes,  is  most  marked  -w  (432). 
on  those  crystals  which  are  characterized  by  the  presence 
of  an  axis  of  symmetry  which  is  at  the  same  time  a 
hemimorphic  axis.  In  order  to  detect  the  presence  of  elec- 
trified surfaces  resulting  from  change  in  temperature  a  mix- 
ture of  red  lead  and  sulphur  may  be  employed.  If  this  be 
shaken  in  a  small  bellows  so  that  the  particles  become  elec- 
trified by  friction  and  blown  into  the  atmosphere  the  particles 
positively  electrified  are  attracted  to  faces  on  the  crystal 
showing  the  negative  sign,  while  the  particles  negatively 
electrified  are  attracted  to  the  faces  with  the  opposite  sign. 
In  this  way  some  of  the  electrified  surfaces  on  the  crystal 
become  red  and  others  become  yellow. 

Projections.  Figure  91  shows  a  gnomonic  projection  paral- 
lel to  001  for  the  mineral  rutile.  The  points  upon  this  pro- 
jection are  distributed  symmetrically  about  the  pole  and  also 
about  four  lines  passing  through  the  pole  and  at  angles  of 
45°  to  one  another.  These  four  lines  of  symmetry  exhibited 


88 


CRYSTALLOGRAPHY 


by  the  projection  correspond  to  the  trace  of  the  two  second- 
ary and  two  intermediate  planes  of  symmetry  characteristic 
of  the  holohedral  class.  The  height  of  the  plane  of  projec- 
tion above  the  centre  of  the  crystal  is  marked  by  the  ground 
circle  whose  radius  corresponds  to  the  height.  It  will  be 
observed  here  that  the  fore  and  aft  pace  is  identical  with  the 


Fig.  91 


Fig.  92 


right  and  left  pace.  This  characteristic  tetragonal  pace  is 
designated  p0  which  in  the  tetragonal  system  is  irrationally 
related  to  r0  —  the  height  of  the  plane  of  projection  above 
the  centre,  here  marked  by  the  radius  of  the  ground  circle. 
Before  it  is  possible  to  write  the  symbols  for  any  of  the  faces 
exhibited  on  this  projection  it  is  necessary  arbitrarily  to 
assign  symbols  to  one  or  more  faces  so  as  to  fix  the  value  of 
p0.  This  having  been  done  the  symbols  for  all  the  faces 
present  can  readily  be  worked  out.  Figure  92  is  a  stereo- 
graphic  projection  for  rutile  exhibiting  the  same  faces  as  are 
shown  on  the  gnomonic  projection. 


CHAPTER   X 
HEXAGONAL   SYSTEM 

In  the  hexagonal  system  are  included  twelve  classes.  Many 
of  the  commonest  substances  crystallize  in  this  system  and 
in  some  respects,  both  on  account  of  the  number  of  substances 
crystallizing  in  it,  and  on  account  of  the  varying  degrees 
of  symmetry  exhibited,  this  must  be  regarded  as  the  most 
important  system. 

Symmetry.  Crystals  of  the  holohedral  class  exhibit  the 
following  symmetry: 

(a)  Axes.     One  hexad  —  three  diad  axes  lying  in  a  plane  at 
right  angles  to  the  hexad  axis.     These  three  axes  make 
with    one   another  angles  of   60°  and    120°.      In    this 
same  plane  three  axes  of  symmetry  alternate  with  the 
three  previously  referred  to.     Three  of  these  axes  are 
secondary  and  three  intermediate  axes  of  symmetry. 
Planes.     One  chief  plane  containing  six  axes  of  symmetry 
-  this  plane  is  at  right  angles  to  the  hexad  axis;   three 
planes,  each  containing  the  hexad  axis  and  one  secondary 
axis  of   symmetry;   three   planes   each    containing   the 
hexad  axis  and  one  intermediate  axis  of  symmetry, 
(c)  Centre  of  symmetry  present. 

These  various  elements  of  symmetry  are  diagrammatically 
illustrated  in  Figures  93  and  94. 


(b) 


,- 


Fig.  94 


Fig.  93 

Standard  Orientation.     Crystals  belonging  to  the  hexagonal 
system   are   by   agreement   oriented  so   that  the  hexad  axis 


90  CRYSTALLOGRAPHY 

is  vertical  while  a  diad  axis  is  right  and  left.  As  there  are 
two  sets  each  containing  three  diad  axes,  there  is  for  each 
substance  crystallizing  in  the  hexagonal  system  a  choice 
as  to  the  set  from  which  the  right  and  left  axis  shall  be 
chosen.  The  crystallographer  who  first  describes  the  crystals 
of  a  substance  arbitrarily  makes  the  choice  and  his  orienta- 
tion of  the  crystal  is  afterwards  strictly  observed.  The 
three  diad  axes  of  symmetry,  one  of  whose  members  occupies 
the  right  and  left  position,  are  known  as  secondary  axes, 
The  three  diad  axes  of  the  other  set  are  intermediate. 

Axes  of  Reference.  Faces  of  all  hexagonal  crystals,  regard- 
less of  the  class  to  which  they  belong,  may  be  referred  to 
three  horizontal  axes  coincident  in  direction  with  the  three 
secondary  axes  of  symmetry  and  one  vertical  axis  which 
coincides  with  the  hexad  axis  of  symmetry.  The  three 
horizontal  axes  of  reference  are  equivalent,  that  is,  any 
plane  which  intersects  two  or  three  of  them  meets  the 
axes  so  as  to  form  intercepts  whose  lengths  are  simply 
and  rationally  related  to  one  another.  Since  these  three 
axes  are  equivalent  they  are  designated  ai,  a2  and  a3  with 
the  peculiar  order  and  distribution  of  signs  exhibited  in 
Figure  95.  The  vertical  axis  designated 
by  the  letter  c  is  unequal  to  the  three 
lateral  axes  since  any  crystal  face  inter- 
secting the  vertical  and  one  or  more 
lateral  axes  cuts  off  on  the  vertical  axis 
a  length  which  is  irrationally  related  to 
the  length  on  any  one  of  the  lateral  axes. 
This  equivalence  of  three  axes  of  refer- 
ence lying  in  the  horizontal  plane  and 
their  inequivalence  to  the  vertical  axis 

c  corresponds  closely  to  the  relationship  observed  in  the  tetra- 
gonal system. 

Classification  of  All  Possible  Planes  Intersecting  Hexagonal 
Axes  of  Reference.  Certain  planes  have  such  directions  as 
to  intersect  the  vertical  axis  at  finite  distances  while  others 
intersect  the  vertical  axis  only  at  infinity,  because  there  lies 
in  these  planes  a  line  which  is  parallel  to  the  vertical  axis. 
These  two  main  subdivisions  may  be  still  further  separated 
as  follows: 


HEXAGONAL    SYSTEM  91 

(a)  The  plane  intersects  vertical  axis: 

1.  Lateral  axes  not  cut, 

2.  Three  lateral  axes  cut  unequally, 

3.  Three  lateral  axes  cut,  two  equally,  one  at  half  the 

distance, 

4.  Two  lateral  axes  cut  equally,  the  third  at  infinity. 
(6)  The  plane  does  not  intersect  the  vertical  axis: 

1.  Three  lateral  axes  cut  unequally, 

2.  Three  lateral  axes  cut,  two  equally,  the  third  at  half 

the  distance, 

3.  Two  lateral  axes  cut  equally,  the  third  at  infinity. 
This  classification  presents  to  us  in  the  hexagonal  system 

seven  possibilities,  each  of  which  corresponds  to  a  type  of 
crystal  form  similar  to  the  types  studied  in  connection  with 
the  previous  two  systems.  The  following  table  gives  the 
names  of  the  forms  in  the  hexagonal  system  and  indicates 
the  nature  of  their  intersection  of  the  axes  of  reference: 

1.  Vertical  axis  cut,  two  lateral  axes  cut  equally,  the  other 

at  infinity  —  pyramid  of  the  first  order, 

2.  Vertical  axis  cut,  two  lateral  axes  cut  equally,  the  other 

at  half  the  distance  —  pyramid  of  the  second  order, 

3.  Vertical    axis    cut,    three   lateral    axes    cut   unequally  — 

dihexagonal  pyramid, 

4.  Vertical  axis  cut,  lateral  axes  not  cut  —  basal  pinacoid, 

5.  Vertical   axis  not   cut,  two   lateral   axes  cut   equally  the 

other  at  infinity  —  prism  of  the  first  order, 

6.  Vertical  axis  not  cut,  two  lateral  axes  cut  equally,  the 

other  at  half  the  distance  —  prism  of  the  second  order, 

7.  Vertical  axis  not  cut,   lateral  axes  cut  unequally  —  di- 

hexagonal prism. 

Hexagonal  Symbols.  In  this  system  there  are  three  equiva- 
lent axes  of  reference  lying  in  one  plane  and  a  fourth  axis  of 
a  different  kind  which  is  normal  to  the  plane  of  these  three. 
Hence  it  will  be  necessary  in  writing  general  symbols  to 
introduce  a  fourth  letter.  For  all  other  systems  only  three 
letters  are  required.  The  fourth  letter  required  for  general 
symbols  of  the  hexagonal  system  is  i  and  stands  in  the  follow- 
ing relationship:  h  >  k  >  i  >  1.  It  will  be  particularly 
noted  that  this  is  not  the  alphabetical  order.  It  would  be 


92  CRYSTALLOGRAPHY 

quite  possible  to  indicate  the  direction  of  hexagonal  planes 
by  using  only  two  indices  for  the  lateral  axes  since  when  the 
indices  of  two  of  the  lateral  axes  is  known  the  third  one  may 
be  calculated.  The  general  relationship  existing  between  the 
lengths  of  the  intercepts  of  the  three  lateral  axes  may  be 
expressed  as  follows:  if  the  shortest  intercept  be  considered 
unity,  the  longest  intercept  n,  then  the  intercept  of  the  third 

axis  is -•      This    may    be    established     by    reference    to 

n  —  1 

Figure  96  as  follows: 

Let  AB  be  a  plane  intersect- 
ing the  three  lateral  hexagonal 
axes  of  reference  in  the  points 
A,  B  and  C,  giving  the  three 
intercepts  AD,  BD  and  CD.  The 
shortest  of  these  CD  =  1;  the 
longest  BD  =  n.  From  C  draw 
Q6  CE  parallel  to  AD.  Then  the 

triangles  ABD  and  CBE  are  simi- 
lar and  AD:  CE::BD:  BE  /.  AD  =  CE'BD        Ln  n 


BE  n-  1        n  -  1 

Since  the  plus  and  minus  signs  are  arranged  alternately 
on  the  axes  of  reference  of  the  hexagonal  system  it  follows 
that  a  plane  which  intersects  the  lateral  axes  necessarily 
intersects  both  positive  and  negative  ends  of  these  axes. 
Keeping  in  mind  this  alternation  of  positive  and  negative 

signs  and  of  the  parameter  ratio  — - — ai:na2:  —  Ia3  we  con- 
n—I 

elude  that  the  algebraical  sum  of  the  indices  for  the  three 
lateral  axes  must  in  all  hexagonal  symbols  be  zero.  This 
may  be  established  by  conversion  of  the  above  general 
parameter  ratios  into  the  index  symbols  of  Miller: 

Parameter  ratio,  — —  ai:  na2:  —  Ia3, 
n  —  1 

Reciprocal  ratio, :-: , 

n       n        1 

Indices,  (n  —  1):  1:  —  n 

In  the  hexagonal  system,  therefore,  when  the  indices  of 
two  lateral  axes  are  known  the  third  may  be  readily  derived 


HEXAGONAL    SYSTEM  93 

by  assigning  to  it  such  a  value  that  the  sum  of  all  three 
indices  becomes  zero.  This  theorem  may  be  employed  to 
check  the  correctness  of  hexagonal  symbols;  thus  symbols 
such  as  the  following  are  manifestly  impossible:  (4311), 

(2232). 

HOLOHEDRAL    CLASS 

Pyramid  of  the  First  Order.  The  faces  of  this  type  inter- 
sect two  lateral  axes  at  equal  distance  from  the  centre  and 
are  parallel  to  the  third.  The  intercept  on  the  vertical  axis, 
in  accordance  with  the  characteristics  of  this  system,  is 
irrationally  related  to  the  intercepts  on  the  lateral  axes. 
There  are  many  possible  forms  of  this  pyramid,  some  steeper 
intersecting  the  vertical  axis  at  a  greater  distance,  others 
flatter  with  a  shorter  intercept  on  this  axis.  The  one  who 
first  describes  a  crystal  of  a  hexagonal  substance  arbitrarily 
selects  one  of  these  pyramids  as  the  unit  or  ground  pyramid. 
Its  symbol,  therefore,  becomes  (1011).  The  ratio  between 
the  lengths  of  the  intercepts  on  the  vertical  and  lateral  axes 
for  this  pyramid  gives  us  the  ratio  c:  a,  which  becomes  a 
characteristic  constant  for  all  crystals  of  this  substance. 
The  value  for  c  may  be  either  greater  or  less  than  that  for  a, 
thus,  for  beryl  — 

c:a:  :  .8643:  1, 
for  corundum  — 

c:a:  :  1.3636:  1. 

In  different  hexagonal  substances  the  ratio  c :  a  varies  but 
it  is  always  irrational.  This  corresponds  to  the  condition  which 
maintains  in  the  tetragonal  system.  The  hex- 
agonal pyramid  of  the  first  order  (Figure  97) 
is  enclosed  by  twelve  faces  each  of  which  in 
the  idealized  form,  where  all  the  faces  occur 
at  equal  distance  from  the  centre,  is  an  isos- 
celes triangle.  There  are  twelve  polar  edges 
and  six  basal  edges.  The  corners  are  of  two 
kinds,  two  hexahedral  and  six  tetrahedral. 
The  symbol  (hOhl)  is  capable  of  representing 
all  possible  pyramids  of  the  first  order.  FiS-  97 

Pyramid  of  the  Second  Order.  The  faces  of  this  type 
intersect  two  lateral  axes  at  equal  distances  from  the  centre, 


94 


CRYSTALLOGRAPHY 


the  third  at  one-half  the  distance.  The  vertical  axis  may  be 
intersected  at  varying  distances,  in  this  way  giving  rise  to 
many  individual  forms,  some  flatter,  some  steeper.  Geomet- 
rically the  pyramid  of  the  second  order  is  similar  to  the  one 


Fig.  98 


Fig.  99 


already  described,  but  differs  from  it  in  orientation  (Figure 
98).  The  relationship  between  these  two  pyramids  is  the 
same  as  that  which  holds  between  the  pyramids  of  the  first 
and  second  orders  of  the  tetragonal  system.  The  symbol 
for  the  unit  pyramid  of  the  second  order  is  (1121),  while 
that  for  all  pyramids  of  this  type  is  (kkhl),  in  which  h  =  2k. 
These  pyramids  of  the  first  and  second  order  are  represented 
in  combination  in  Figure  99. 

Dihexagonal   Pyramid.     Each   face   of   this   type    cuts    all 
three  of  the  lateral  axes  at  different  distances  from  the  centre 


Fig.  100 


Fig.  101 


and  the  vertical  axis  at  a  distance  which  is  irrationally 
related  to  the  lengths  on  the  lateral  axes.  The  symmetry  of 
this  class  requires  twenty-four  faces  of  this  kind  to  constitute 
the  crystal  form.  Each  face  is  a  scalene  triangle.  There  are 


HEXAGONAL    SYSTEM 


95 


twelve  basal  edges  and  twenty-four  polar  edges  —  two  sets 
of  twelve  each.  Two  corners  representing  the  highest  and 
lowest  points  of  the  crystal  are  dodecahedral,  while  there  are 
twelve  tetrahedral  corners,  two  sets  of  six  each  marking  the 
points  of  exit  of  the  secondary  and  intermediate  axes  of 
symmetry.  This  is  the  most  general  form  of  the  hexagonal  sys- 
tem, with  the  general  symbol  (kihl)  (Figure  100).  The  dihex- 
agonal  pyramids,  more  frequently  observed  on  the  mineral 
beryl,  are  the  following:  (2133),  (5165),  (5494),  (2131),  (3141), 
(5161)  and  (4261). 

The  mode  of  intersection  on  the  lateral  axes  of  the  three 
pyramids  is  shown  in  Figure  101. 

Hexagonal  Prisms.  In  this  system,  as  in  the  tetragonal 
system,  there  are  three  prisms  corresponding  in  cross-section 
to  the  three  pyramids.  They  are  designated  - 

1.  Prism  of  the  first  order  with  the  symbol  (1010), 

2.  Prism  of  the  second  order  with  the  symbol  (1120), 

3.  Dihexagonal  prism  with  the  symbol  (kihO). 

The  first  and  second  of  these  prisms  are  hexagonal  in  cross- 
section  and  differ  from  one  another  only  in  orientation.  Each 
of  these  prisms  is  bounded  by  six  planes  meeting  in  six  edges 
parallel  to  the  vertical  axis.  The  cross-section  of  the  dihex- 
agonal  prism  is  twelve-sided.  A  dihexagonal  prism  in  which 
all  twelve  angles  are  equal  would  involve  irrationality  of  the 
axial  parameters,  and  consequently  is  not  crystallographically 
possible.  This  form  is  bounded  by  twelve  planes  meeting 
in  twelve  edges  parallel  to  the  vertical  axis.  All  these  prisms 
are  open  forms  and  consequently  occur  on  crystals  only  in 
combination  with  one  or  more  of  the  pyramids  or  with  the 
basal  pinacoid.  Figures  102,  103  and  104  represent  the 
three  prisms  in  combination  with  the  next  form. 


x 

| 

lo^i 

1 

1 

4 

I 

. 

1 

^r 

1 

i 
i 

N 

—  -l~f- 

—  k 

Fig.  102 


Fig.  103 


Fig.  104 


96 


CRYSTALLOGRAPHY 


Basal  Pinacoid.  This  form  consists  of  two*  faces  parallel 
to  one  another  occurring  at  opposite  ends  of  the  vertical 
axis.  Symbol  (0001).  This  is  an  open  form  which  may  occur 
on  crystals  in  combination  with  prisms  and  pyramids. 

Limiting  Forms.  The  limitation  of  the  various  forms  in 
the  hexagonal  system  is  in  most  respects  closely  related  to 
that  already  observed  for  the  tetragonal  system.  Certain 
pyramids  are  steeper  than  the  unit  pyramid,  others  flatter. 
The  former  may  be  supposed  to  find  their  limit  in  the  prism, 
the  infinitely  steep  pyramid,  and  the  latter  in  the  pinacoid, 
the  infinitely  flat  pyramid.  With  regard  to  the  intercepts 
on  the  lateral  axes  variations  may  reach  their  limits  in  two 
ways:  first  k  may  increase  in  value  until  it  is  equal  to  h 
when  the  pyramid  or  prism  of  the  first  order  results;  second, 
k  may  diminish  in  value  until  it  is  equal  to  one-half  h,  so 
giving  rise  to  the  pyramid  or  prism  of  the  second  order. 
The  relationships  of  the  pyramids  to  the  prisms  and  basal 
pinacoid  and  of  the  dihexagonal  forms  to  the  forms  of  the 
first  and  second  order  may  be  diagrammatically  expressed  in 


(0001) 


(hOhl) 


(1010). 


-   (kihl)^ 

I 

-(kihO)  - 
Fig.  105 


-(kkhl) 

t 
-  (1120J 


Fig.  106  —  Beryl  showing  _the 
prism  of  the  first  order,  m,  (10lO); 
the  basal  pinacoid,  c,  (0001);  the 
pyramids  of  the  first  order,  u,  (2021) 
and  p,  (1011);  the  pyramid  of  the 
second  order,  s,  (1121)  and  the  di- 
hexagonal pyramid  v,  (2131). 

Figure   105.     Beryl   (Figure   106),   pyrrhotite  and  langbanite 
crystallize  in  the  holohedral  hexagonal  class. 

Calculation  of  the  Ratio  c :  a.  This  can  be  most  simply 
calculated  in  the  hexagonal  system  from  the  basal  angle 
between  two  faces  of  the  unit  pyramid  of  the  second  order 


HEXAGONAL    SYSTEM 


97 


(1122).  When  this  is  known  the  value  of  c  (a  being  unity) 
is  the  tangent  of  half  the  inner  angle.  From  various  other 
angular  measurements  this  ratio  may  be  determined  although 
not  so  simply.  The  relationship  between  the  pace  distance 
Po  and  c  is  expressed  as  follows: 

Po  =  f  c. 

Projections.  Figure  107  represents  a  gnomonic  projection 
of  beryl  as  shown  in  Figure  106.  An  examination  of  this 
projection  indicates  that  the  points  are  symmetrically  dis- 
tributed with  regard  to  six  lines  of  symmetry  and  that  these 
lines  pass  through  the  pole  which  is  a  point  of  hexad  symme- 
try. The  pace  lengths  measured  in  the  direction  of  the  three 
lines  of  symmetry,  which  correspond  to  the  axes  of  reference 


Fig.  107 


Fig.  108 


ai,  a2,  a3,  are  equivalent  for  all  three  directions  and  are  con- 
sequently designated  p0.     The  length  p0  is  irrationally  related 
to  r0,  whose  magnitude  is  represented  by  the  radius  of  the 
ground  circle.     Figure   108  is  a  stereographic  projection  of 
the  forms  shown  in  the  gnomonic  projection   (Figure   107). 


HOLOHEDRAL    HEMIMORPHIC    CLASS 

This  class  may  be  easily  derived  from  the  holohedral  class 
by  dropping  the  chief  plane  of  symmetry.  This  means  the 
simultaneous  disappearance  of  the  centre  and  of  the  six  diad 
axes.  Although  the  hexad  axis  remains,  the  forms  which 
meet  one  end  of  this  axis  do  not  necessarily  intersect  the 
opposite  end  in  a  similar  fashion.  The  symmetry  of  this 


98 


CRYSTALLOGRAPHY 


class  is  therefore  one  hemimorphic  hexad  axis,  three  second- 
ary planes  of  symmetry  and  three  intermediate  planes.  On 
crystals  of  this  class  the  three  prisms  characteristic  of  holohe- 
drism  occur  but  may  be  distinguished  from  the  corresponding 
holohedral  forms  by  means  of  etching  figures.  The  other 
four  forms,  viz.,  basal  pinacoid,  dihexagonal  pyramid  and 
the  pyramids  of  the  first  and  second  orders  are  each  divided 

into  two  forms  designated  respectively 

upper    and    lower,    with    the   following 

symbols : 

!._  Pyramid  of  the  first  order,  upper 

(hOhl), 

2.  Pyramid  of  the  first  order,  lower 

(hOhl), 
3^  Pyramid  of  the  second  order,  upper 

(kkhl), 

4.  Pyramid  of  the  second  order,  lower 
(kkhl), 

5.  Dihexagonal  pyramid,  upper  (kihl), 

6.  Dihexagonal  pyramid,  lower  (kihl), 

7.  Basal  pinacoid,  upper  (0001), 
(OOuT)                                    8.  Basal  pinacoid,  lower  (OOOl). 

The  mineral  zincite  which  belongs  to  this  class  is  illustrated 
in  Figure  109. 


Fig.  109— Zincite  show- 
ing the  prism_  of  the  first 
order,  m,  (1010);  the  up- 
per half  pyramid  of  the 
first  order,  p,  (lOll)  and 
the  lower  basal  pinacoid, 


RHOMBOHEDRAL  HEMIHEDRAL  CLASS 

Rhombohedral  hemihedrism  results  from  the  extension  of 
the  faces  contained  in  alternate  dodecants  with  the  suppres- 
sion of  those  faces  occurring  in  the  other  dodecants.  The 


Fig.  110 


HEXAGONAL    SYSTEM 


99 


only  forms  of  the  hexagonal  system  which  produce  geometri- 
cally new  forms  by  this  method  of  selection  are  those  whose 
faces  are  confined  within  a  single  dodecant  —  the  pyramid  of 
the  first  order  and  the  dihexagonal  pyramid.  Crystals  of 


111 


this  class  are  characterized  by  the  presence  of  one  triad 
axis  and  three  diad  axes,  three  intermediate  planes  and  a 
centre  of  symmetry.  This  method  of  hemihedral  selection 
is  shown  in  Figures  110  and  111. 

Rhombohedron.  The  application  of  this  method  of  selec- 
tion to  the  pyramid  of  the  first  order  (Figure  110)  gives  rise 
to  a  pair  of  rhombohedra  —  congruent  forms  —  each  of  which 
is  bounded  by  six  faces.  The  faces  are  all  rhombs  in  outline. 
There  are  six  polar  edges  and  six  basal  edges.  In  certain 


Fig.  112  —  The    cube    as   the    intermediate   form   between   flat    and    steep 

rhombohedra. 

rhombohedra  the  angle  over  the  polar  is  greater  than  that 
over  the  basal  edge;  in  others  the  opposite  relationship 
holds.  In  the  flat  rhombohedra  the  longer  diagonal  of  the 


100 


CRYSTALLOGRAPHY 


rhomb  is  horizontal;  in  the  steep  rhombohedra  this  position 
is  occupied  by  the  shorter  diagonal.  This  type  of  hemihe- 
drism  being  indicated  by  the  letter  K,  the  symbols  for  these 
two  types  of  rhombohedra  are: 

1.  Positive  rhombohedron  —  symbol  of  the  unit  form 

K(lOll),  general  symbol  /c(hOhl), 

2.  Negative  rhombohedron  —  symbol  of  the  unit  form 

/c(OlTl),  general  symbol  /c(Ohhl). 

A  rhombohedron  whose  basal  and  polar  angles  are  equal 
to  each  other  would  be  geometrically  identical  with  the  cube. 
This  may  be  regarded  as  the  limiting  form  between  flat  and 
steep  rhombohedra  (Figure  112). 

Scalenohedron.  The  application  of  this  method  of  selec- 
tion to  the  dihexagonal  pyramid  (Figure  111)  gives  rise  to 
two  congruent  forms  bounded  by  twelve  faces,  each  face  in 
outline  a  scalene  triangle.  There  is  a  great  variety  of  scale- 
nohedra,  certain  of  them  being  very  flat  while  others  are 
steep  and  sharp.  Not  only  is  there  a  variation  in  the  inter- 
cept on  the  vertical  axis  but  there  is  also  variation  in  the 
ratios  of  intercepts  on  the  lateral  axes.  The  following  are 
the  symbols  for  the  various  scalenohedra: 

Positive  scalenohedron,    /c(kihl), 
Negative  scalenohedron,  K(ikhl). 

The  six  planes,  whose  directions  are  indicated  by  the  ad- 
jacent pairs  of  basal  edges  of  the  scalenohedron,  if  extended 


Fig.  113  Fig.  114  — Calcite  showing  the 

prism  of  the  first  order,  m,  (1010) 
and  the  negative  rhombohedron,  e, 
K  (0112). 

to  enclose  space  give  rise  to  a  rhombohedron,  known  as  the 
rhombohedron  of  the  middle  edges.     All  those  scalenohedra, 


HEXAGONAL   SYSTEM'  •' 


which  intersect  the  lateral  axes  in  the  same  fashion  but 
differ  from  one  another  in  varying  lengths  on  the  vertical 
axis,  have  basal  edges  identical  in  direction  and  are  some- 
times referred  to  as  derivatives  of  the  same  rhombohedron 
of  the  middle  edges.  (Figure  113.) 

A  very  large  number  of  minerals  crystallize  in  this  class, 
some   of   the  commonest   of  which   are   as   follows:     calcite, 


Fig.  115  — Calcite 
showing  the  positive 
rhombohedron,  r, 
/c(10ll)  and  the  posi- 
tive scalenohedron,  v, 
K  (2131). 


Fig.  116  — Calcite 
showing  the  prism  of 
the  first  order,  m, 
(lOlO);  the  positive 
rhombohedron,  r, 
K(lOll)  and  the  sca- 
lenohedron, v,  «(2131). 


Fig.  117  — Hematite 
showing  the  positive 
rhombohedra,  t, 
*c(3035),_r,  K(1011)  and 
u,  K(1014);  the  posi- 
tive scalenohedron,  i, 
«(4265)  and  the  pyra- 
mid of  the  second 
order  n,  (2243). 


CaC03    (Figures    114,  115    and    116),   siderite,   FeCO3  corun- 
dum, A12O3,  and  hematite,  Fe2O3  (Figure  117). 

TRAPEZOHEDRAL  HEMIHEDRAL  CLASS 

The  selection  and  extension  of  one-half  the  faces  within  the 
dodecant  and  the  regular  alternation  of  the  selection  in  adja- 
cent dodecants  gives  rise  to  trapezohedral  forms.  Only  the 
dihexagonal  pyramid  is  capable  of  yielding  geometrically 
new  forms  by  this  method  of  selection.  Figure  118  shows 
the  method  of  selection  and  the  nature  of  the  new  forms 
resulting  from  it.  These  hexagonal  trapezohedra  are  enan- 
tiomorphous  and  bear  the  general  symbols: 

Right  hexagonal  trapezohedron,  r(kihl), 
Left  hexagonal  trapezohedron,     r(hikl). 
Crystals  belonging  to  this  class  are  devoid  of  centre  and 


CRYSTALLOGRAPHY 


Fig.  118 

planes  of  symmetry  but  exhibit  the  axes  of  symmetry  which 
characterize  the  holohedral  class. 


PYRAMIDAL  HEMIHEDRAL  CLASS 

The  forms  peculiar  to  this  class  of  hemihedrism  may  be 
obtained  by  selecting  and  extending  faces  of  the  holohedra 
according  to  the  method  already  indicated  for  the  tetragonal 
pyramidal  class.  Only  the  dihexagonal  pyramid  and  prism 
produce  new  forms  according  to  this  method  of  selection. 
Forms  of  this  class  are  symmetrical  about  the  chief  plane, 
the  hexad  axis  and  the  centre. 

Pyramid  of  the  Third  Order.  From  the  dihexagonal  pyramid 
by  this  method  of  selection  a  congruent  pair  of  pyramids  of 


Fig.  119 

the  third  order  is  obtained  (Figure  119).  These  pyramids  are 
geometrically  identical  with  the  pyramids  of  the  first  and 
second  orders  but  differ  from  them  in  orientation.  Symbols: 


HEXAGONAL    SYSTEM 


103 


Positive  pyramid  of  the  third  order,    Tr(kihl), 
Negative  pyramid  of  the  third  order,  Tr(hikl). 


Fig.  121 

The  relationship  of  the  pyramids  of  the  three  orders  to 
one  another  is  shown  in  Figure  120  by  the  way  in  which  their 
faces  intersect  the  axes  of  reference.  The  heavy  lines  repre- 
sent the  pyramid  of  the  third  order. 

Prism  of  the  Third  Order.     From  the  dihexagonal  prism 
are  obtained  the  positive  and  negative  prisms  of  the  third 
order.    Their  relationship  to  one  another  and  to  the  prism  from 
which  they  are  derived  is  indicated  in  Figure  121.     Symbols: 
Positive  prism  of  the  third  order,    Tr(kihO), 
Negative  prism  of  the  third  order,  Tr(hfkO). 


Fig.  122  —  Apatite  showing  the  three 
prisms  —  first  order,  ra,  ( lOlO)  ;_second  order,  a, 
(1120)  and  third  order,  h,  TT (2 130);  three  pyra- 
mids oj  the  first  order,  y,  (2021),  x,  (loll)  and 
r,  (1012);  the  pyramid  of  the  second  order, 
s,  (1121.),  the  pyramid  of  the  third  order, 
M,  7r(2131)  and  the  basal  pinacoid,  c,  (0001). 

The  minerals  of  the  apatite  group  exhibit  this  class  of 
symmetry.  Figure  122  represents  combinations  of  the  com- 
mon form  of  apatite,  3Ca3(PO4)2.  Ca(F,Cl)2. 


104 


CRYSTALLOGRAPHY 


PYRAMIDAL  HEMIHEDRAL  HEMIMORPHIC  CLASS 

This  class  whose  only  symmetry  is  that  about  a  hemimor- 
phic  hexad  axis  may  be  regarded  as  a  hemimorphic  derivative 
of  the  last  class.  The  crystal  forms  are  the  prisms  of  the 
first,  second  and  third  orders  which  are  geometrically  identi- 
cal with  those  of  the  pyramidal  hemihedral  class  and  the 
basal  pinacoid  and  pyramids  of  the  first,  second  and  third 
orders,  but  in  this  class  these  four  forms  are  broken  each 
into  two  forms  designated  upper  and  lower.  The  symbols 
for  these  eight  forms  are  as  follows: 

Upper  pyramid  of  the  first  order  —  (hOhl), 
Lower  pyramid  of  the  first  order  —  (hOhl), 
Upper  pyramid  of  the  second  order  —  (kkhl), 
Lower  pyramid  of  the  second  order  —  (kkhl), 

Positive  upper  pyramid  of  the  third 
order  —  (kihl), 

Positive  lower  pyramid  of  the  third 
order  —  (kihl), 

Negative  upper  pyramid  of  the  third 
order  —  (hikl), 

Negative  lower  pyramid  of  the  third 
order  —  (hikl), 

Upper  basal  pinacoid  —  (0001), 
Lower  basal  pinacoid  —  (0001). 
Fig.    123  —  Strontium       The  mineral  nepheline  has  been  placed 
antimonyl  dextro-tartrate   m  thig  clagg  ag  &  regult  of  examination  of 
showing  the  prism  of  the      .    ,  .        r  e  ,  ,    .      ,. 

first  order,  m,  (1010);  the  etchmS  fiSures  formed  °n  certam  faces' 
upper  hemipyramid  of  the  Strontium  antimonyl  dextro-tartrate, 
first  order,  o,  (lOl'l)  and  S^SbO^^C^Oe^  belongs,  according 
the  lower  hemipyramid  of  to  the  goniometric  measurements  by 
the  first  order,  x,  (2021).  Traube>  to  this  t  of  symmetry.  The 

corresponding  lead  compound  gives  rise  to  crystals  of  the 
same  symmetry  (Figure  123). 


TRAPEZOHEDRAL  TETARTOHEDRAL  CLASS 

Crystals  of  this  class  are  characterized  by  the  presence  of 
one  triad  axis  of  symmetry  and  three  hemimorphic  diad 
axes.  The  characteristic  tetartohedral  forms  may  be  derived 
by  the  simultaneous  application  of  the  methods  of  selection 


HEXAGONAL    SYSTEM 


105 


by    which    the    rhombohedral    and    trapezohedral    classes    of 
hemihedrism  were  obtained.     This  selection  as  applied  to  the 


Fig.  124 

dihexagonal    pyramid   is   shown   in    Figure    124.     The   forms 
peculiar  to  this  class  are  briefly  referred  to  below: 

Trigonal  Trapezohedron.  From  the  dihexagonal  pyramid 
are  obtained  the  two  geometrically  new  forms  illustrated  in 
Figure  124.  These  forms  show  the  right  and  left  handed 
characteristics  already  observed  in  the  hexagonal  trapezohe- 
dra.  They  are  enantiomorphous.  In  addition  to  these  two 
other  forms  are  obtained,  one  identical  with  each  of  those 
illustrated  in  Figure  124,  but  differing  from  it  only  in  orien- 
tation. These  four  crystallographic  forms  therefore  constitute 
two  pairs  of  congruent  forms.  They  are  designated  as  follows: 

1.  Positive  right  trigonal  trapezohedron  Kr(kihl), 

2.  Negative  right  trigonal  trapezohedron  Kr(hkil), 

3.  Positive  left  trigonal  trapezohedron  Kr(hikl), 

4.  Negative  left  trigonal  trapezohedron  Kr(khil). 


hikO 


Fig.  125 


106 


CRYSTALLOGRAPHY 


Ditrigonal  Prism.  If  this  method  of  selection  be  applied 
to  the  dihexagonal  prism  two  new  forms  are  obtained,  dif- 
fering from  one  another  only  in  orientation  (Figure  125). 
These  are  designated: 

1.  Right  ditrigonal  prism  —  /cr(kihO), 

2.  Left  ditrigonal  prism      — /cr(hikO). 

Trigonal  Pyramid.  From  the  pyramid  of  the  second  order 
a  pair  of  congruent  forms  is  obtained.  They  are  known  as: 


lie 


1120 


Fig.  126 


Fig.  127 — Quartz  showing  the  prism  of  the  first 
order,  ra,  (lOlO)  and  the  positive  and  negative 
rhombohedra  of  the  first  order  simulating  the  pyr- 
amid of  the  first  order,  r,  /c(10Tl)  and z,  K  (Olll). 


-  KT(kkhl), 
Kr(hkkl). 


1.  Right  trigonal  pyramid,  second  order 

2.  Left  trigonal  pyramid,  second  order  — 

Trigonal  Prism.     The  same  method  of  selection  applied  to 
the  prism  of  the  second  order  produces  a  pair  of  congruent 


Fig.  128  and  129  —  Quartz:  the  former  left  and  the  latter  right  handed. 
The  forms  common  to  the  two  crystals  are  the  prism  of  the  first  order,  m 
(1010);  the  positive  rhombohedron,  r,  /c(1011);  the  negative  rhombohedron, 
z,  K(0lll);  and  the  trigonal  pyramid,_s,  /cr(2Ul).  On  Figure  128  the  left 
positive  trigonal  trapezohedron,  x,  Kr(6151);  on  Figure  129  the  right  positive 
trigonal  trapezohedron,  x,  /cr(5161). 


HEXAGONAL    SYSTEM  107 

forms  known  as  trigonal  prisms.  Their  relation  to  one  another 
and  to  the  prism  of  the  second  order  from  which  they  are 
obtained  is  shown  in  Figure  126.  Symbols: 

Right  trigonal  prism  —  /cr(1120), 

Left  trigonal  prism   -  — /cr(2110). 

The  mineral  quartz,  SiO2,  shows  the  symmetry  of  this 
class  although  geometrically  the  faces 
present  are  usually  so  arranged  as 
to  suggest  a  higher  symmetry  (Figure 
127).  The  occurrence,  however,  of 
the  trigonal  pyramids  and  trapezo- 
hedra  is  frequent,  and  from  their 
position  upon  the  crystals  it  is  easy  Fig.  130 — Cinnabar  show- 
to  determine  whether  the  crystal  is  inS  the  prism  of  the  first 
right  or  left.  The  faces  are  usually  order>  m>  (101°);  the  basal 

small   and   lie  almost  in  the  plane  of  pmatcold'    <'     0™?''    the 
f  positive    rhombohedron,    r, 

the  prismatic  face  and  when  above  K(loll).  the  negative  rhom- 
the  plane  are  either  to  the  right  or  bohedra,  n,  *(0221)  and  g, 
left,  as  illustrated  in  Figures  128  and  «(0112);  the  left  positive 
129,  the  former  of  which  is  left-  trigonal  trapezohedron,  /, 
handed  and  the  other  right-handed. 
Cinnabar,  HgS  also  crystallizes  in  this  class.  (Figure  130.) 

RHOMBOHEDRAL  TETARTOHEDRAL  CLASS 

In  this  class  the  crystals  possess  a  triad  axis  and  a  centre 
of  symmetry.  The  characteristic  forms  are  such  as  might 
be  obtained  by  the  simultaneous  application  of  the  rhombohe- 
dral  and  pyramidal  methods  of  hemihedral  selection  to  cer- 
tain holohedra  of  the  hexagonal  system.  The  only  holohedra 
which  give  rise  to  new  forms  are  the  three  pyramids  and  the 
dihexagonal  prism. 

Rhombohedron  of  the  First  Order.  The  application  of 
this  method  of  selection  to  the  pyramid  of  the  first  order 
produces  the  rhombohedra  of  the  first  order  (Figure  110). 
While  geometrically  identical  with  the  rhombohedra  pre- 
viously described  they  differ  from  them  physically. 

Rhombohedron  of  the  Second  Order.  By  applying  this 
method  of  tetartohedral  selection  to  the  pyramid  of  the 
second  order  a  pair  of  rhombohedra  is  obtained  geometri- 


108 


CRYSTALLOGRAPHY 


cally  similar  to  those  just  described,  but  differing  from  them 
in  orientation.  Figure  131  shows  the  relation  of  these  two 
forms  to  the  pyramid  of  the  second  order.  Symbols^ 

1.  Positive  rhombohedron  of  the  second  order  K-n-(kkhl), 

2.  Negative  rhombohedron  of  the  second  order  o-(hkkl). 


Fig.  131 

Rhombohedron  of  the  Third  Order.  Two  pairs  of  rhom- 
bohedra  geometrically  similar  to  those  of  the  first  and  second 
orders  are  obtained  by  this  method  from  the  dihexagonal 
pyramid.  The  derivation  of  two  of  these  forms  from  the 
holohedral  form  is  shown  in  Figure  132.  The  symbols  for 
the  four  rhombohedra  of  the  third  order  are  as  follows: 


Fig.  132 

1.  Positive  right  rhombohedron  of  the  third  order  —  KTr(kihl), 

2.  Positive  left  rhombohedron  of  the  third  order  —  KTr(hikl), 

3.  Negative  right  rhombohedron  of  the  third   order  —  o-(ihkl), 

4.  Negative  left  rhombohedron  of  the  third  order  —  o-(ikhl). 
Prism  of  the  Third   Order.     From  the  dihexagonal  prism 

two  prisms  of  the  third  order  are  obtained.     They  are  geo- 
metrically identical   with  the  prisms  obtained  by  pyramidal 


HEXAGONAL    SYSTEM 


109 


hemihedrism.  The  minerals  dioptase,  H2CuSiO4,  (Figure 
133),  phenacite  (Figure  134),  ilmenite  and  dolomite,  (Ca,Mg)- 
CO3  (Figure  135)  belong  to  this  class. 

The  prisms  of  the  first  and  second  orders  and  the  basal 
pinacoid  do  not  change  their  form  by  the  application  of  this 


Fig.  133  —  Diop- 
tase showing  the 
prism  of  the  second 
order,  a,  (1120);  the 
negative  rhombo- 
hedron  of  the  first 


order, 


c(0221) 


and  the  rhombo- 
hedron  of  the  third 
order,  x,  o-(1341). 


Fig.  134  —  Phenacite  show- 
ing the  prism  of  the  second 
order,  a,  (1120);  the  positive 
rhombohedra  of  the  first  order, 
z,  /c(10ll)andd,  K(10l2);  the 
negative  rhombohedron  of  the 
first  order,  r,  /c(0lll);  the 
pyramid  of  the  second  order, 
p,  (1123);  the  rhombohedron 
of  the  third  order,  x,  *7r(2132) ; 
the  rhombohedron  of  the  third 
order,  s,  K7r(2131). 


Fig.  135  —  Dolomite 
showing  the  prism  of  _the 
second  order,  a,  (1120); 
the  basal  pinacoid,  c, 
(0001) ;  the  positive  rhom- 
bohedron of  the  first 
order,  r,  /c(10ll);  the  posi- 
tive rhombohedron  of  the 
third  order,  y,  KIT  (3251). 


method  of  selection.  They  occur  on  crystals  of  this  class 
but  are  identical  geometrically  with  those  of  the  holohedral 
class. 

TRIGONAL  HEMIHEDRAL  CLASS 

The  symmetry  of  this  class  consists  of  a  triad  axis,  three 
hemimorphic  diad  axes,  the  chief  plane  and  three  intermediate 
planes  of  symmetry.  The  forms  peculiar  to  this  class  may 
be  obtained  by  selecting  one-half  the  faces  from  the  following 
holohedral  forms:  prism  of  the  first  order,  dihexagonal  prism, 
pyramid  of  the  first  order  and  dihexagonal  pyramid. 

In  this  method  of  selection  the  faces  occurring  in  alternate 
dodecants  above  are  extended  while  those  in  the  three  dode- 
cants  immediately  below  are  extended  at  the  same  time. 
There  are  positive  and  negative  forms. 


110 


CRYSTALLOGRAPHY 


Trigonal  Pyramid.     Two  congruent  forms  are  obtained  from 
the  pyramid  of  the  first  order  by  the  selection  indicated  in 


Fig.  136 

Figure    136.     They    differ   from    the   trigonal    pyramids    pre- 
viously described  principally  in  orientation. 

1.  Positive  trigonal  pyramid  of  the  first  order  —  (hOhl), 

2.  Negative  trigonal  pyramid  of  the  first  order  —  (Ohhl). 


ono 


Fig.  137 

Trigonal  Prism.  The  selection  by  this  method  of  one-half 
the  faces  of  the  prism  of  the  first  order  (Figure  137)  gives 
rise  to  two  congruent  forms: 

Positive  trigonal  prism  of  the  first  order   -  -  (1010), 
Negative  trigonal  prism  of  the  first  order  —  (OHO). 


Fig.  138 


HEXAGONAL    SYSTEM 


111 


Ditrigonal  Pyramid.  These  forms  are  obtained  from  the 
dihexagonal  pyramid.  Their  relations  to  one  another,  and 
to  the  parent  form  are  indicated  in  Figure  138. 


Fig.  139 

Positive  di  trigonal  pyramid   -  -  (hikl), 
Negative  ditrigonal  pyramid  —  (ikhl). 
Ditrigonal  Prism.     Figure  139  indi- 
cates   the    relationship    of   these  two 
prisms  to  one  another  and  to  the  di- 
hexagonal prism. 

Positive  ditrigonal  prism  —  (hikO), 

Negative  ditrigonal  prism  —  (ikhO.) 

Most    of    the   forms   belonging    to 

this    class    are    geometrically    similar    ing  the  trigonal  prisms  of  the 

to   certain   forms   belonging  to   other    firsl  order>  m  (101°)  and  * 

classes,     but     differ     from     them     in    (^!°);  ^hexagonal  prism 

of  the  second  order,  o,  (1120)  ; 
orientation.  the  trigonal  pyramidg  of  the 

Until  recently  no  substance  had  first  order,  p,  (1011);  •*-, 
been  discovered  crystallizing  in  this  (0111);  e,  (Oll2);  and  the 
class,  but  in  1909  Rogers  and  Pa-  hexag°rial  pyramid  of  _the 
lache  called  attention  to  the  sym- 
metry  of  the  new  mineral  benitoite 
as  a  first  representative  of  this  symmetry  (Figure  140). 


Fig.  140  —  Benitoite  show- 


_ 
(2241)* 


TRIGONAL  HEMIHEDRAL  HEMIMORPHIC  CLASS 

This  class  is  symmetrical  about  one  hemimorphic  triad 
axis  and  about  three  intermediate  planes.  It  may  be  con- 
ceived as  a  hemimorphic  representative  of  the  class  just 
described.  If  from  the  last  class  the  principal  plane  of  sym- 


112 


CRYSTALLOGRAPHY 


metry  disappear  and  with  it  the 
three  diad  axes  we  obtain  the 
degree  of  symmetry  characteristic 
of  this  class.  Its  characteristic 
forms  are  therefore  the  same  as 
those  of  the  preceding  class  except 
that  the  basal  pinacoid  and  the 
various  pyramidal  forms  are  here 
each  represented  by  upper  and  lower 

Fig.  141  —  Tourmaline  show-  forms. 

ing  the  trigonal  prism  of  the  first  The  mineral  tourmaline,  a  corn- 
order,  m  (Olio);  the  prism  of  plex  silicate,  is  the  best  known  rep- 
the  second  order,  a,  (1120);  the  resentative  of  this  type  of  sym- 
upper  half  forms  of  the  follow-  ^-  ,  .,  ., 

,   ,    ,  metry.      Figure    141    exhibits    the 
ing:  the  positive  rhombohedron 

of  the  first  order,  r,  /c(10ll);  forms  most  frequently  observed 
the  jpositive  scalenohedron,  t,  on  crystals  of  this  mineral.  Its 
*c(2131);  the  negative  rhombohe-  hemimorphism  about  the  vertical 
dron  of  the  first  order,  O,K  (0221);  axis  ig  welj  marked.  Like  mogt 
at  the  lower  end  the  sole  terminal  ,  ,.  .  ,  .,  , 

form  is  the  positive  rhombohe-  hemimorphic  minerals  it  becomes 
dron,  T,  /c(OlTI).  strongly  pyro-electnc  with  change 

of    temperature  and  is  commonly 
used  for  the  demonstration  of  this  phenomenon. 


TRIGONAL  TETARTOHEDRAL  CLASS 

Crystals  belonging  to  this  class  possess  a  chief  plane  and  a 
triad  axis  of  symmetry.  The  characteristic  forms  consist  of 
trigonal  pyramids  and  prisms  of  three  different  orders  which 
may  be  derived  by  selecting  one-quarter  of  the  faces  of  the 
various  pyramids  and  prisms  possible  in  the  holohedral 


Fig.  142 

hexagonal  class.     Those  forms  derived  from  the  pyramid  and 
prism  of  the  first  order  are  known  as  trigonal  pyramids  and 


HEXAGONAL    SYSTEM 


113 


prisms  of  the  first  order.  The  pyramid  and  prism  of  the 
second  order  give  rise  to  trigonal  pyramids  and  prisms  of  the 
second  order,  while  from  the  dihexagonal  pyramids  and 
prisms  are  obtained  the  corresponding  trigonal  pyramid  and 
prism  of  the  third  order.  The  method  of  selection  of  one- 
quarter  of  the  faces  of  the  holohedra  is  shown  in  Figures 
142  and  143.  Since  no  substance  has  been  discovered  up  to 


Fig.  143 


the  present  time  crystallizing  in  this  class  the  development 
of  the  forms  and  their  symbols  is  here  omitted. 


TRIGONAL  TETARTOHEDRAL  HEMIMORPHIC  CLASS 

Crystals  of  this  class  possess  only  a  triad  hemimorphic 
axis  of  symmetry  coinciding  with 
the  vertical  axis.  This  class  may 
therefore  be  regarded  as  the  hemi- 
morphic representative  of  the  pre- 
ceding class.  The  basal  pinacoid 
and  the  trigonal  pyramids  of  the 
first,  second  and  third  orders  are 
in  this  class  each  divided  into  a  Fig.  144  —  Sodium  periodate 
pair  of  forms  —  upper  and  lower.  —  a  plan  showing  the  positive 
The  trigonal  prisms  of  the  three  trig°nal  Pyramid  of  the  first 

orders    may    occur    in    the    same  °rder'  ['  (1011)/,  thfe  "egaf  ? 

J  trigonal    pyramid    of    the    first 

geometrical    form    as    in    the    last  order>  ^   (0^1);    the  trigonal 

class,   although  the  hemimorphism   pyramid  of  the  third  order,  t, 
of    the    axis    of    symmetry    might    (2134).    The  negative  end  of  the 

be  determined   by  etching    figures  vertical.  axis  is  cut_by  a  larse 

,  T    ',   .   ..  c<     T          basal  pinacoid  (0001). 

or     by     pyro-electricity.      Sodium 

periodate,  NaIO4  +  3  H2O,  which  crystallizes  in  this  class  is 
shown  in  Figure  144. 


CHAPTER  XI 

RHOMBIC    SYSTEM 

In  the  rhombic  system  are  included  three  different  classes 
each  of  which  has  a  degree  of  symmetry  peculiar  to  itself. 
Crystals  belonging  to  these  various  classes  are  nevertheless 
closely  related  to  one  another  in  the  following  manner: 

1.  The    forms    characteristic    of    these    classes    can    all    be 

referred  to  three  inequivalent  axes  at  right  angles  to 
one  another, 

2.  The  forms  characteristic  of  those  classes  of  lower  sym- 

metry may  be  derived  by  regular  selection  of  one-half 
the  faces  characteristic  of  some  form  belonging  to  the 
holohedral  class, 

3.  There   are   certain   holohedra   belonging   to   the   highest 

class  which  do  not  produce  geometrically  new  forms 
by  hemihedral  or  hemimorphic  selection.  These  forms 
occur  in  combination  with  the  characteristic  forms  in 
each  of  the  classes  belonging  to  the  rhombic  system. 
Such  forms  are  the  rhombic  prism  and  pinacoids. 

HOLOHEDRAL  CLASS 

Symmetry.  Crystals  of  this  class  are  symmetrical  about 
three  diad  axes  at  right  angles  to  one  another,  about  three 
planes,  each  of  which  contains  two  axes  of  symmetry,  and 
about  a  centre.  The  three  axes,  while  possessing  the  same 
degree  of  symmetry  are  not  equivalent,  as  each  penetrates 
the  crystal  in  a  different  fashion. 

Standard  Orientation.  Rhombic  crystals  are  so  oriented 
that  the  three  diad  axes  are  respectively  fore  and  aft,  right 
and  left  and  vertical.  There  are  six  possible  ways  of  orient- 
ing a  crystal  so  as  to  be  in  accord  with  this  general  restric- 
tion. The  one  who  first  describes  a  crystal  of  a  substance 


RHOMBIC    SYSTEM  115 

belonging  to  the  rhombic  system  usually  places  it  so  that  the 
diad  axis  corresponding  to  the  greatest  length  of  the  crystal 
has  the  vertical  position.  In  determining  the  direction  of 
the  other  two  axes,  account  is  taken  of  the  form  of  the  cross- 
section  of  the  crystal.  In  this  system,  as  the  name  suggests, 
the  rhombus  is  the  common  cross-section  of  crystals.  When 
the  vertical  direction  has  been  selected  that,  axis  of  sym- 
metry which  corresponds  to  the  shorter  diagonal  of  the  rhomb, 
in  the  form  most  prominently  developed,  is  placed  fore  and 
aft,  while  the  longer  diagonal  of  the  rhomb  is  right  and  left. 
It  sometimes  happens  however  that  the  crystals  from  which 
the  substance  was  first  described  do  not  represent  the  ordi- 
nary development  of  the  substance.  Later  it  may  be  found 
that  the  axis  which  was  selected  for  the  vertical  direction  in 
reality  usually  corresponds  to  the  shortest  dimension  of  the 
crystal,  or  that  rhomb  which  was  most  prominent  in  the 
cross-section  and  consequently  decisive  in  indicating  which 
of  the  other  two  axes  should  be  right  and  left  and  fore  and 
aft,  does  not  as  a  rule  correspond  to  faces  of  large  develop- 
ment. Regardless  of  these  later  discoveries  the  orientation 
given  to  the  crystal  by  the  original  investigator  is  retained, 
so  that  in  certain  rhombic  crystals,  as  pictured  in  the  text 
books,  the  vertical  direction  really  corresponds  to  its  shortest 
dimension.  This  is  often  observed  on  crystals  of  barite, 
BaSO4,  celestite,  SrSO4,  and  anglesite,  PbSO4. 

Axes   of   Reference.     The   axial   cross   for   crystals   of   the 
rhombic  system  consists  of  three  lines  at  right  angles  to  one 
another,  coinciding  with  the  directions  of 
the  three  axes    of   symmetry.     A  crystal 
face  which  intersects  any  two  of  these  axes 
gives  rise  to  intercepts  that  are  irrationally 

related  to  one  another.     Owing  to  the  in-    _b 

equivalence  of  the  axes  they  are  designated 
by  different  letters,  a,  b  and  c,  as  illus- 
trated in  Figure  145. 

Ground  Form.     Before  progress  can  be 
made  in  assigning  symbols  to  most  of  the  pjg  145 

faces  which  occur   upon  the    crystal  it  is 
necessary  arbitrarily  to   select  some  one  form  each  of  whose 
faces  intersects  the  three  axes,  and  to  agree  that  the  ratios 


116  CRYSTALLOGRAPHY 

cut  off  on  these  axes  by  a  face  of  this  form  shall  be  the  funda- 
mental ratios  of  the  three  axes. 

This  ratio  is  further  simplified  by  expressing  the  values 
a  and  c  in  terms  of  b,  which  is  unity.  These  values  are 
known  as  axial  ratios  and  are  characteristic  for  each  sub- 
stance crystallizing  in  the  rhombic  system.  Since  the  hori- 
zontal section  through  the  ground  form  is  a  rhomb  with  the 
greater  diagonal  right  and  left,  it  follows  that  the  length  of 
the  right  and  left  intercept  is  always  greater  than  that  on  the 
fore  and  aft  axis,  so  that  of  the  axial  ratios  a  is  always  less 
than  b.  In  accord  with  this  the  fore  and  aft  axis  is  known 
as  the  brachyaxis,  while  the  other  lateral  axis  is  the  macro- 
axis.  This  is  sometimes  indicated  on  the  letters  by  signs, 
thus  a,  b,  c.  For  topaz  the  values  are  - 

a:b:c:  :  .5285:  1:  .9539. 

From  these  axial  ratios  the  values  for  the  polar  elements 
p0:  qo:  r0  may  be  readily  obtained  by  the  following  transforma- 
tion: 

c 

Po  =  -;  qo  =  c;  r0  ==  1. 
a 

Classification  of  All  Planes  with  Regard  to  Intersection  of 
Rhombic  Axes  of  Reference. 

(a)  Plane  intersects  only  one  axis: 

(1)  a  axis  cut  —  macropinacoid, 

(2)  b  axis  cut  —  brachypinacoid, 

(3)  c  axis  cut  —  basal  pinacoid; 

(6)  Plane  intersects  two  axes: 

(1)  a  axis  not  cut  —  brachydome, 

(2)  b  axis  not  cut  —  macrodome, 

(3)  c  axis  not  cut  —  rhombic  prism; 

(c)  The  plane  cuts  all  three  axes  —  rhombic  pyramid. 

Macropinacoid.  This  form,  like  all  pinacoids,  consists 
of  two  planes  parallel  to  one  another.  It  is  an  open  form 
and  can  only  occur  on  crystals  in  combination  with  other 
forms.  Its  symbol  is  (100). 

Brachypinacoid.  There  are  here  two  planes  parallel  to  one 
another.  The  centre  of  symmetry  requires  these  planes  to 
occur  in  pairs.  It  is  an  open  form.  Symbol  (010). 


RHOMBIC  SYSTEM 


117 


Basal  Pinacoid.  This  form  is  similar  to  the  two  forms 
already  described,  except  that  its  planes  are  horizontal  and 
intersect  the  vertical  axis.  Symbol  (001). 

The  arrangement  for  these  three  pinacoids  is  indicated  in 
Figure  146,  exhibiting  the  three  pinacoids  in  combination  with 
one  another.  The  pinacoids  are  invari- 
able and  so  stand  in  contrast  to  the  four 
other  types,  each  of  which  comprehends 
a  considerable  number  of  distinct  forms. 

Brachydome.  A  face  intersecting  the 
b  and  c  axes  at  finite  distances  will  in 
this  system  necessarily  be  accompanied 
by  three  other  faces.  The  brachydome 
is  an  open  form  and  cannot  occur  in- 
dependently, but  is  frequently  observed 
in  combination  with  others.  The  four 


Fig.  146 


planes  meet  in  four  edges  parallel  to  the  a  axis.  The  brachy- 
dome occurs  in  many  different  forms  according  to  the  ratios 
at  which  the  b  and  c  axes  are  intersected.  The  unit  brachy- 
dome has  the  symbol  (Oil).  The  natter  and  steeper  domes 
may  be  indicated  by  the  symbols  (Okh)  and  Ohk)  respectively. 
(Figure  147.) 

Macrodome.  This  form  in  many  respects  resembles  the 
brachydome,  except  that  its  four  planes  intersect  in  edges 
parallel  to  the  b  axis.  Each  of  these  planes  cuts  the  a  and  c 
axes  at  finite  distances.  (Figure  148.)  The  unit  macrodome 


Fig.  147 


Fig.  148 


has  the  symbol  (101),  while  the  flatter  and  steeper  domes  have 
the  symbol  (kOh)  and  (hOk)  respectively.  In  the  rhombic 
system,  pinacoids  and  domes  are  named  from  the  lateral 


118 


CRYSTALLOGRAPHY 


axis    which    they    do    not    intersect,    thus:     the    brachydome 
intersects  the  macro  axis  but  does  not  cut  the  brachy  axis. 

Rhombic  Prism.  This  form  closely  resembles  the  two 
forms  just  described  but  differs  from  them  in  that  the  four 
faces  meet  in  edges  parallel  to  the  vertical  axis,  while  each 
face  of  the  prism  intersects  the  two  lateral  axes  (Figure  149). 


Fig.  149 


Fig.  150 


Symbols   (110)   for  the  unit  prism  and   (hkO)   and  (khO)  are 
comprehensive  symbols  for  all  other  prisms. 

Rhombic  Pyramid.  The  rhombic  pyramid  is  bounded  by 
eight  planes  each  of  which  is  a  scalene  triangle  (Figure  150). 
Its  twelve  edges  are  of  three  types,  four  in  each  set.  The 
six  tetrahedral  corners  are  of  three  pairs,  one  pair  corre- 
sponding to  the  points  of  exit  of  each  diad  axis  of  symmetry. 
The  unit  pyramid  known  as  the  ground  form  intersects  the 
three  axes  at  ratios  which  become  standard  for  each  rhombic 
substance.  The  general  symbol  for  the  rhombic  pyramid  is 
(hkl).  There  is  a  series  of  pyramids  intersecting  the  lateral 
axes  in  the  same  ratio  as  the  ground  form  but  with  varying 
intercepts  on  the  vertical  axis.  These  are  known  as  unit 
pyramids.  Another  series,  known  as  the  brachypyramids, 
intersects  the  a  axis  at  varying  lengths  while  a  third  series, 
the  macropyramids,  intersects  the  b  axis  at  different  lengths. 
The  symbols  for  these  three  series  are  indicated  as  follows: 

Unit  pyramids       -  (hhk)  and  (kkh), 

Brachypyramids  —  (khh)  and  (hkk), 

Macropyramids  --  (hkh)  and  (khk). 

Limiting  Forms.  Three  of  the  seven  forms  characteristic 
of  the  rhombic  system  are  invariable.  These  are  the  pina- 
coids.  Each  of  the  other  four  types  includes  a  considerable 


RHOMBIC  SYSTEM 


119 


number  of  possible  forms.  This  is  implied  by  the  use  of  the 
letters  h,  k  and  1  in  writing  their  symbols. 

The  pyramid  is  the  most  general  type  in  this  system  be- 
cause by  giving  limiting  values  to  the  various  indices  all 
seven  forms  of  the  rhombic  system  may  be  derived.  When 
the  intercept  of  the  pyramid  on  the  vertical  axis  becomes 
infinitely  long  a  prism  results,  while  the  basal  pinacoid  is 
obtained  if  the  intercept  on  the  vertical  axis  be  infinitely 
short.  Similarly  by  varying  the  lengths  of  the  fore  and  aft 
axis  the  series  of  pyramids  culminates  in  the  brachydome 
and  macropinacoid.  When  the  pyramid  is  given  its  limiting 
values  on  the  right  and  left  axis  the  two  forms  obtained  are 
the  macrodome  and  the  brachypinacoid. 

The  prisms  and  domes  are  also  variable  forms,  each  of  them 
swinging  between  two  pinacoidal  limits.  Thus  from  the 
prism  the  macropinacoid  may  be  obtained  by  making  the 
intercept  on  the  b  axis  infinity  and  the  brachypinacoid  by 
making  the  intercept  on  the  b  axis 
zero.  In  the  same  way  the  series  of 
macrodomes  swings  between  the  limits 
indicated  by  the  macropinacoid  and 
the  base,  while  the  brachydome  finds 
its  limits  in  the  brachypinacoid  and 
the  base.  These  relationships  are  dia- 
grammatically  indicated  in  Figure  151. 

Calculation  of  the  Ratios  a :  b :  c.    The 
value  of  c  can  be  most  simply  calculated 

from  the  inner  basal  angle  between  the  faces  (Oil)  and 
(Oil).  Similarly  a  may  be  derived  from  the  angle  be- 
tween the  faces  (110)  and  (iTO).  The  information  con- 
cerning the  ratios  is  complete  when  we  know  a:  b  and  b:  c. 
While  the  above  represents  the  simplest  calculation  it  is  not 
the  one  ordinarily  followed.  The  derivation  of  the  ratios 
a:b:c  from  other  angles  usually  involves  spherical  trigono- 
metry. 

Gnomonic  Projection.  Figure  4  presents  a  gnomonic  pro- 
jection for  some  of  the  forms  commonly  observed  on  the 
mineral  topaz.  The  plane  of  projection  is  parallel  to  the 
basal  pinacoid.  It  may  be  observed  that  the  projection  points 
are  distributed  symmetrically  about  two  lines  at  right  angles 


hkO 


100 


010 


120  CRYSTALLOGRAPHY 

to  one  another,  and  about  a  central  point  —  the  pole  of  the 
projection.  The  two  lines  of  symmetry  in  a  projection  corre- 
spond to  the  two  planes  of  symmetry  in  the  rhombic  system 
which  stand  at  right  angles  to  the  basal  pinacoid,  while  the 
centre  of  symmetry  exhibited  in  the  projection  corresponds 
to  the  diad  axis  which  is  normal  to  the  plane  of  projection. 
In  general  the  elements  of  symmetry  exhibited  in  the  gno- 
monic  projection  are  one  degree  lower  than  the  corresponding 
elements  observed  upon  the  crystals  themselves,  that  is,  the 
planes  and  axes  of  symmetry  become  lines  and  centres  of 
symmetry  in  the  gnomonic  projection. 

The  radius  of  the  circle  on  the  projection,  known  as  the 
ground  circle,  indicates  the  height  of  the  plane  of  projec- 
tion above  the  centre  of  the  crystal.  Since  the  distance  of 
the  various  points  from  one  another  is  dependent  upon 
the  height  of  the  plane  of  projection  it  is  usual  to  indicate  the 
scale  of  a  projection  by  drawing  a  circle  whose  radius  is  the 
height.  On  measuring  the  distance  of  the  various  projec- 
tion points  from  the  right  and  left  line  of  symmetry  it  will 
be  found  that  these  distances  are  all  simply  related  to  one 
another.  The  same  relationship  holds  with  regard  to  the 
distance  of  the  points  from  the  fore  and  aft  line  of  symmetry. 
If  we  select  a  fore  and  aft  pace  unit  p0  and  a  right  and  left 
pace  unit  q0  it  will  be  found  that  po,  qo  and  r0  —  the  height 
of  the  plane  of  projection  above  the  centre  as  shown  by  the 
radius  of  the  ground  circle  —  are  for  this  system  all  irra- 
tionally related  to  one  another.  The  method  by  which  the 
symbols  and  crystallographic  constants  can  be  graphically 
determined  from  a  gnomonic  projection  has  been  already 
explained.  In  studying  the  projection  of  a  rhombic  crystal, 
not  hitherto  described,  it  is  necessary  to  select  arbitrarily 
the  pace  lengths  p0  and  q0  after  which  the  symbols  for  all 
the  faces  may  be  readily  indicated  and  by  graphic  determina- 
tion the  ratios  po,  qo  and  TO  can  be  worked  out.  The  trans- 
formation of  these  pace  lengths  or  polar  elements  into  the 
axial  ratios  a:  b:  c  is  shown  in  the  following: 

c 

Po  =  -;  qo  =  c;  r0  =  1. 
a 

For  the  mineral  topaz  the  constants  as  expressed  in  the  two 
forms  are  as  follows: 


RHOMBIC    SYSTEM 


121 


a:b:  c:  :  .5285:  1:  .9539, 
p0:q0:r0:  :  1.8049:  .9539:  1. 

Figure  14  is  a  stereographic  projection  representing  for 
topaz  the  forms  shown  in  the  gnomonic  projection. 

Combinations.  In  this  system  six  of  the  seven  types  are 
represented  by  groups  of  faces  which  do  not  enclose  space. 
All  rhombic  forms,  except  the  pyramids,  appear  on  crystals 
only  in  combination  with  other  forms.  The  faces  of  the 
pinacoids  do  not  even  intersect  to  give  rise  to  edges.  Figures 

h 


Fig.  152  —  Sulphur  representing 
the  unit  pyramid,  p,  (111);  a  second 
pyramid,  s,  (113);  the  basal  pina- 
coid,  c,  (001);  and  the  brachydome, 
n,  (Oil). 


Fig.  153 — Hypersthene  showing  the 
macropinacoid,  a,  (100);  the  brachy- 
pinacoid,  6,  (010) ;  the  unit  prism,  m, 
(110);  the  prism,  n,  (120);  the  brachy- 
dome, h,  (014)  and  four  pyramids:  e, 
(212);  o,  (111);  u,  (232);  i,  (211). 


5,  152  and  153  present  combinations  of  rhombic  forms  occur- 
ring on  the  minerals  topaz,  sulphur  and  hypersthene. 

Mineral  Examples.  In  addition  to  the  minerals  already 
mentioned  as  crystallizing  in  this  class,  the  following  are 
important:  staurolite,  chalcocite  and  olivine. 


SPHENOIDAL  CLASS 

There  are  two  classes  of  lower  symmetry  occurring  in  the 
rhombic  system,  one  of  which  is  hemihedral,  the  other  hemi- 
morphic.  Of  the  three  types  of  hemihedral  selection  which 
have  been  applied  to  the  earlier  systems,  only  one  is  capable 
of  producing  new  geometrical  forms  in  the  rhombic  system. 
This  selection  consists  in  the  extension  of  those  faces  occurring 
in  one-half  of  the  octants  and  the  suppression  of  the  faces 
occurring  in  the  alternate  octants.  As  has  been  already 
indicated  this  type  of  hemihedrism,  known  as  tetrahedral  in 


122 


CRYSTALLOGRAPHY 


the  cubic  system,  sphenoidal  in  the  tetragonal  system  and 
rhombohedral  in  the  hexagonal  system,  can  only  give  new 
forms  when  applied  to  those  holohedra  whose  faces  are 
confined  to  a  single  octant.  In  the  rhombic  system  this  is 
true  for  the  pyramids  only  and  consequently  the  only  hemihe- 
dral  type  characteristic  of  this  class  consists  of  a  pair  of 


Fig.  154 

enantiomorphous  forms,  one  right,  the  other  left.  They 
have  a  general  resemblance  to  the  sphenoids  of  the  tetragonal 
system.  Each  form  is  enclosed  by  four  scalene  triangles. 
Their  derivation  from  the  rhombic  pyramid  is  shown  in  Fig- 
ure 154.  The  letter  K  indicates  this  type  of  hemihedrism 
while  the  symbols  for  the  two  forms  are  as  follows: 

Positive  sphenoid  —  /c(hkl), 
Negative  sphenoid  —  Ac(hkl). 
Symmetry  about  three  diad  axes  at 
right  angles  to  one  another  is  charac- 
teristic of  this  class. 

These  forms  are  usually  observed 
in  combination  with  one  or    more  of 
the  holohedral  forms  of  the  rhombic 
system.      Figure  155  shows  the  occur- 
rence of  the  right  sphenoid  in  corn- 
Fig.  155 —  Magnesium  sul-  bination  with  the  rhombic  prism  as 
phate,  MgSO4  +  7H2O,  show-  exhibited    on     magnesium     sulphate, 
ingtheprism,m,(110)andthe    M   go    +7HQ      andon     z^     ^\- 
positive  sphenoid,  o,  /c(lll).      ,&       *  '  . 

phate,     ZnSO4  +  7  H2O.      Cream    of 

tartar,  rochelle  salt  and  tartar  emetic  also  crystallize  in  this 
class.  It  is  not  unusual  for  both  the  sphenoids  to  appear  on 
the  same  crystal,  in  this  way  simulating  the  rhombic  pyramid. 


RHOMBIC  SYSTEM 


123 


When  this  occurs,  however,  there  is  usually  a  marked  differ- 
ence in  central  distance  between  the  right  and  left  forms,  while 
the  faces  on  the  one  form  may  be  rough  the  other  smooth. 


HEMIMORPHIC    CLASS 

A  number  of  substances  crystallize  in  such  a  way  as  to 
suggest  the  holohedral  rhombic  class  when  only  one  end  of  the 
crystal  is  examined  but  on  closer  investigation  it  is  found 
that  they  are  characterized  by  two  planes  of  symmetry  and 
only  one  diad  axis.  The  faces  intersecting  this  diad  axis  at 
opposite  ends  are  frequently  different  from  one  another. 
These  forms  are  said  to  be  hemimorphic  about  the  diad  axis 
and  the  axis  itself  is  a  hemimorphic  axis.  It  is  usual  to 
orient  crystals  of  this  class,  where  only  one  axis  of  symmetry 
is  present,  so  that  this  axis  is  vertical.  The  crystal  forms 
are  as  follows: 

(a)  Rhombic  prism,  brachypinacoid,  macropinacoid,  all 
apparently  holohedral, 

(6)  Pyramid,  macrodome  and  brachydome  and  basal 
pinacoid. 

Each  of  the  forms  in  the  latter  group  becomes  two  forms  in 
the  hemimorphic  class  which  are  designated  upper  and  lower, 


Fig.  156  —  Calamine  showing  the 
unit  prism,  m,  (110);  the  macropina- 
coid,  a,  (100);  the  brachypinacoid,  b, 
(010) ;  the  half  forms  of  the  following : 
upper  basal  pinacoid,  c,  (001);  the 
macrodomes,  t,  (301)  ands,  (101);  the 
brachy domes,  i,  (031);  e,  (Oil).  The 
lower  end  of  the  crystal  is  enclosed  by 
the  half  pyramid,  v,  (121). 


Fig.  157  —  Struvite  showing  the 
brachypinacoid,  6,  (010);  the  half 
brachydomes,  k,  (041)  and  d,  (Oil); 
the  macrodomes,  s,  (101)  and  Si, 
(101);  the  half  basal  pinacoid,  c, 
(001). 


while  this  difference  is  indicated  in  their  symbols  by  the  sign 
characteristic  of  the  last  index. 


124  CRYSTALLOGRAPHY 

The  best  known  example  of  this  class  is  calamine, 
H2Zn2SiO5,  which  on  account  of  its  characteristic  develop- 
ment is  also  known  as  hemimorphite  (Figure  156).  Struvite, 
NH4MgPO4  +  6  H2O,  also  crystallizes  in  this  class  (Figure 
157).  As  has  already  been  pointed  out  hemimorphic  crystals 
are  ordinarily  pyro-electric.  That  end  of  the  crystal  which 
becomes  negatively  electrified  with  falling  temperature  is 
known  as  the  analogue  pole,  while  the  opposite  end  of  the 
hemimorphic  axis  is  known  as  the  antilogue  pole.1 

1  Some  crystallographers  use  the  variable  index  1  in  the  last  three  sys- 
tems only  for  reference  to  the  vertical  axis.  In  this  case  the  relative 
values  for  the  letters  are  h>k  while  the  value  of  1  is  not  fixed.  In  this 
work  with  the  exception  of  Figure  151  the  usage  of  the  cubic  system  has 
been  applied  to  the  last  three  systems. 


CHAPTER  XII 
MONOCLINIC   SYSTEM 

There  are  included  in  the  monoclinic  system  three  classes 
of  crystals  each  of  which  differs  from  the  others  in  symmetry. 
The  forms  characteristic  of  each  of  these  classes  may  be  all 
referred  to  the  same  type  of  axes  of  reference.  Those  forms 
which  are  characteristic  of  the  classes  of  lower  symmetry 
may  be  derived  from  certain  of  the  forms  contained  in  the 
holohedral  class  by  regular  selection  of  one-half  the  faces  of 
some  holohedron.  Moreover  there  are  certain  forms  which 
are  common  to  the  various  classes  contained  within  this 
system.  The  low  degree  of  symmetry  which  characterizes 
these  classes  goes  hand  in  hand  as  a  general  rule  with  molec- 
ular complexity.  To  this  system  belongs  a  large  proportion 
of  the  complex  silicates  occurring  in  nature  and  of  the  organic 
products  resulting  from  laboratory  experiment. 

HOLOHEDRAL  CLASS 

Symmetry.  Crystals  of  this  class  are  marked  by  the 
presence  of  one  plane  of  symmetry,  a  diad  axis  at  right 
angles  to  this  plane  and  a  centre.  Four  is  the  largest  number 
of  faces  which  this  low  degree  of  symmetry  requires  in  any 
monoclinic  crystal  form.  This  stands  in  sharp  contrast  to 
the  hexoctahedron  of  the  cubic  system,  whose  multiplicity 
of  faces  is  intimately  connected  with  the  unusually  high 
symmetry  of  the  holohedral  class  of  that  system. 

Standard  Orientation.  It  is  usual  to  orient  monoclinic 
crystals  in  such  a  way  that  the  diad  axis  of  symmetry  is 
right  and  left.  The  one  who  first  describes  the  crystals  of  a 
monoclinic  substance  orients  the  crystal  so  that  a  prominent 
series  of  zone  edges  is  vertical.  There  is  only  one  other 
point  that  must  be  arbitrarily  settled  before  the  orientation 


126 


CRYSTALLOGRAPHY 


of  the  monoclinic  crystal  is  fixed.     It  is  necessary  to  decide 
which  should  be  regarded  as  the  front  of  the  crystal. 

£  Axes  of  Reference.     Monoclinic  forms 

are  referred  to  three  inequivalent  axes. 
The  right  and  left  axis  b  coincides 
with  the  axis  of  symmetry.  The  ver- 
— b  tical  axis  c  which  is  at  right  angles  to 
b  is  usually  parallel  to  some  prominent 
series  of  zone  edges.  The  fore  and  aft 
axis  a  is  at  right  angles  to  b  but  is 
oblique  with  regard  to  c.  The  direc- 
tion which  the  original  investigator 


Fig.  158 


selects  for  the  a  axis  is  usually  prominently  marked  upon 
the  crystal  by  a  series  of  zone  edges  fore  and  aft  in  their 
general  direction  but  sloping  downward  toward  the  front. 
Figure  158  shows  the  relationship  of  these  axes  to  one  an- 
other with  the  letters  and  signs  at- 
tached. Figure  159  represents  a 
crystal  of  orthoclase  on  which  the 
zone  edges  selected  for  the  vertical 
position  are  indicated  by  heavy 
lines,  while  the  edges  whose  direc- 
tion was  selected  for  the  fore  and 
aft  axis  a  are  indicated  by  dotted  Fig.  159— Orthoclase.  The 
lines.  In  this  system  the  three  axes  heavy  vertical  zone  lines  show 
are  designated  by  terms  which  have  tne  direction  chosen  to  repre- 
reference  to  the  angles  which  these  sent  the  vertical  axis,  c;  the 
i  . . ,  , ,  dotted  lines  represent  the  di- 

axes  make  with  one  another.  rection  of  the  oblique  ^  a 

(1)  a=  clmo  axis,   which   is  some-    while  the  axis  b  corresponds 
times  indicated  by  an  oblique    to  the  normal  to  the  face  b. 
sign  above  the  letter  —  a.  Forms  shown  on  the  crystal; 

(2)  b  =  ortho  axis,  to  denote  which    m>  <£®'>  b»  <?10>;  c'  (001)>' 
,,  ,  ,.     .     .,      .      n,  (021);   y,  (201);  x,  (101); 
the  sign  of  perpendicularity  is    0  nu\ 

sometimes    placed   above    the 
letter  — b. 

(3)  c  =  vertical   axis,   indicated   by   a   vertical  sign   placed 
above  the  letter  —  c. 

In  this,  as  in  the  rhombic  system,  the  names  applied  to  the 
axes  are  of  importance  in  the  naming  of  the  various  domes 
and  pinacoids  characteristic  of  the  system. 


MONOCLINIC  SYSTEM 


127 


Classification  of  All  Possible  Planes  with  Reference  to 
Intersection  of  Monoclinic  Axes. 

(a)  Plane  intersects  only  one  axis: 

(1)  a  axis  cut  —  orthopinacoid, 

(2)  b  axis  cut  —  clinopinacoid, 

(3)  c  axis  cut  —  basal  pinacoid, 

(b)  Plane  intersects  two  axes: 

(1)  a  axis  not  cut  —  clinodome, 

(2)  c  axis  not  cut  —  monoclinic  prism, 

(3)  b  axis  not  cut  —  two  possibilities  — 

I  —  Plane  lies  opposite  obtuse  angle  /?  —  negative  hemi-or- 

thodome, 

II  —  Plane    lies  opposite  acute  angle  /3  —  positive  hemi-or- 
thodome, 

(c)  Plane  intersects  three  axes: 

(1)  Plane  opposite  obtuse  angle  /3  —  negative  hemi-pyramid, 

(2)  Plane  opposite  acute  angle  ft  —  positive  hemi-pyramid. 

In  all  the  holohedral  classes  so  far  considered  there  were 
seven,  and  only  seven,  types,  but  in  the  monoclinic  system 
there  are  nine  possible  types.  In  the  triclinic  system,  where 
the  symmetry  is  still  lower,  the  number  is  further  in- 
creased. 

Orthopinacoid.  This  form  consists  of  two  planes  parallel 
to  one  another  and  corresponds  to  the  macropinacoid  of  the 
rhombic  system.  The  faces  intersect  the  fore  and  aft  axis 
a  and  are  parallel  to  the  plane  containing  the  b 
and  c  axes.  Symbol  (100). 

Clinopinacoid.  This  form  is  composed  of  two 
planes  which  are  parallel  to  the  plane  of  sym- 
metry. Symbol  (010).  The  corresponding  form 
in  the  last  system  is  the  brachypinacoid. 

Basal  Pinacoid.  Like  the  other  two  pinacoids 
this  is  an  open  form  consisting  of  a  pair  of 
parallel  planes.  The  c  axis  is  intersected  at  a 
finite  distance,  the  other  axes  at  infinity.  Symbol 
(001). 


Fig  160 


Figure  160  shows  the  three  pinacoids  in  combination  on 
a  crystal  of  pyroxene.  It  may  be  noted  that  these  three 
pinacoids  meet  to  form  angles  as  follows: 


128 


CRYSTALLOGRAPHY 


(100)  and  (010)  —  right  angles, 
(001)  and  (010)  —  right  angles, 

(100)  and  (001)  —  two  acute  and  two  obtuse  angles. 
Clinodome.  Forms  of  this  type  consist  of  four  planes  each 
intersecting  the  b  and  c  axes  at  finite  distances  and  the  a  axis 
at  infinity.  In  the  monoclinic,  as  in  the  rhombic  system, 
some  form  whose  faces  intersect  all  three  axes  is  selected  as 
the  ground  form  and  the  ratios  of  the  lengths  of  the  inter- 
cepts cut  off  on  the  three  axes  are  constants  characteristic 
of  each  crystallized  monoclinic  substance.  For  gypsum, 
CaSO4  +  2  H20,  these  values  are: 

a:b:c:  :  .6895:1:  .4133. 

That  clinodome  which  intersects  the  b  and  c  axes  in  these 
standard  ratios  is  known  as  the  unit  clinodome  and  has  the 
symbol  (Oil).  (Okh)  represents  in  a  gen- 
eral way  all  those  clinodomes  intersecting 
the  vertical  axis  at  a  ratio  less  than  the 
unit  dome.  The  steeper  forms  may  be  rep- 
resented by  (Ohk).  The  edges  of  inter- 
section of  these  four  planes  are  parallel  to 
the  clino  axis.  (Figure  161). 

Monoclinic  Prism.  The  monoclinic  prism 
consists  of  four  planes  intersecting  the  two 
lateral  axes  and  parallel  to  the  vertical  axis.  That  form  which 
intersects  the  a  and  b  axes  in  the  ratios  characteristic  for 
the  substance  is  called  the  unit  prism  and  has  the  symbol 
(110).  Those  prisms  intersecting  the  b 
axis  at  a  relatively  greater  distance  can 
all  be  represented  by  the  general  symbol 
(hkO),  while  those  intersecting  it  at  a  ratio 
less  than  the  unit  prism  are  contained  in 
the  symbol  (khO).  The  four  zone  edges  of 
this  prism  are  parallel  to  the  vertical  axis 
c.  (Figure  162). 

Negative  Hemi-orthodome.  The  faces 
of  this  form  intersect  the  a  and  c  axes  but 
are  parallel  to  the  b  axis.  It  consists  of 
a  pair  of  planes  lying  opposite  the  obtuse  angles  between 
a  and  c.  The  two  faces  for  the  unit  form  may  be  symbo- 
lized 101  and  10T.  In  addition  to  the  unit  form  there 


Fig.  162 


MONOCLINIC   SYSTEM 


129 


are  steeper  and  flatter  negative  hemi-orthodomes  which  may 
be  symbolized  as  (hOk)  and  (kOh)  respectively.  The  posi- 
tion of  this  pair  of  planes  is  indicated  in  Figure  163. 

Positive    Hemi-orthodome.      These    forms    are    similar    to 
the  ones  just  described  except  that  the  planes  lie  opposite  the 


Fig.  163 


Fig.  164 


acute  angles  between  the  a  and  c  axes.  The  symbols  for  the 
two  planes  of  the  unit  form  are  T01  and  10L  Flatter  and 
steeper  positive  hemi-orthodomes  may  be  represented  by 
the  symbols  (kOh)  and  (hOk)  respectively.  (Figure  164). 

Negative  Hemi-pyramid.  This  form  consists  of  four  planes 
intersecting  all  three  axes  and  lying  opposite  the  obtuse 
angle  0.  Their  arrangement  is  shown  in  Figure  165.  The 


Fig.  166 


unit  pyramid  has  the  symbol  (111),  while  (hkl)  is  a  general 
symbol  for  all  negative  hemi-pyramids. 

Positive  Hemi-pyramid.  This  is  in  all  respects  similar  to 
the  form  just  described  except  that_its  planes  lie  opposite 
the  acute  angle  /3.  General  symbol  (hkl)  (Figure  166). 


130 


CRYSTALLOGRAPHY 


Combinations.  In  the  monoclinic  system,  as  a  result  of 
the  low  degree  of  symmetry,  none  of  the  forms  is  represented 
by  a  sufficient  number  of  faces  to  enclose  space.  They  are 
all  open  forms  and  can  only  occur  on  crystals  in  combination 
with  one  another.  Figures  159,  167  and  168  represent  crys- 


Fig.  167  —  Realgar  showing  the 
unit  prism,  m,  (110);  the  prism,  I, 
(210);  the  clinopinacoid,  b,  (010);  the 
basal  pinacoid,  c,  (001);  the  clino- 
dome,  r,  (012);  the  positive  hemi- 
pyramid,  e,  (111). 


Fig.  168  — Ferrous  sulphate,  FeS04 
+  7H2O,  showing  the  unit  prism,  m, 
(110);  the  clinopinacoid,  6,  (010);  the 
basal  pinacoid,  c,  (001);  the  negative 
hemiorthodome,  v,  (101);  the  posi- 
tive hemiorthodome,  t,  (101);  the 
clinodome,  o,  (Oil)  and  the  negative 
hemipyramid,  r,  (111). 


tals  of  orthoclase,  KAlSi3O8,  realgar,  AsS,  and  ferrous  sul- 
phate, FeSO4  +  6  H2O,  on  which  these  various  forms  occur 
in  combination. 

Limiting  Forms.  The  relationship  of  the  forms  of  the 
monoclinic  system  to  one  another  is  in  most  respects  similar 
to  that  indicated  as  characteristic  of  the  rhombic  system. 
A  pyramid  by  varying  the  length  of  the  vertical  axis  reaches 
its  limits  in  the  basal  pinacoid  on  the  one  hand  and  a  mono- 
clinic  prism  on  the  other.  By  varying  the  intercept  of  the 
fore  and  aft  axis  the  pyramids  swing  between  the  limits 
indicated  by  the  orthopinacoid  and  the  clinodome.  A  similar 
variation  of  the  right  and  left  axis  produces  a  series  of  pyra- 
mids which  find  their  limits  in  the  clinopinacoid  and  the 
orthodome.  When  we  say  that  the  pyramid  (hkl)  is  the  most 
general  form  in  this  system  we  mean  thereby  that  all  the 
other  forms  of  the  system  may  be  derived  from  it  merely  by 
giving  the  limiting  values  to  its  various  indices.  The  domes 
and  prisms  being  variable  forms  swing  between  the  limit  set 
by  a  pair  of  pinacoids,  thus,  the  prism  between  the  ortho- 
pinacoid and  the  clinopinacoid,  the  orthodomes  between  the 


MONOCLINIC   SYSTEM 


131 


orthopinacoid   and  the  base  and  the  clinodome  between  the 
clinopinacoid  and  the  base. 

Calculation  of  Monoclinic  Ratios  a:b:c  and  0.  In  this, 
as  in  the  rhombic  system,  the  ratios  of  the  a  and  c  axes  are 
expressed  in  terms  of  b  which  is  unity.  The  angle  between 
the  a  and  c  axes  is  oblique  in  this  system  and  varies  con- 
siderably in  different  monoclinic  substances.  This  angle  is 
indicated  by  the  Greek  letter  0.  It  is  necessary,  therefore,  to 
determine  for  monoclinic  crystals  the  magnitude  of  the  angle 
j8  in  addition  to  the  axial  ratios  a:  b:  c. 

The  intersections  of  the  a  and  c  axes  give  rise  to  two 
oblique  angles,  one  greater  and  the  other  less  than  90°.  It 
is  usual  to  employ  the  letter  /3  to  indicate  the  smaller  of  these. 
This  angle  may  be  determined  by  direct  measurement  of  the 
angle  between  the  orthopinacoid  and  the  base. 

Gnomonic  Projection.  Figure  169  is  a  gnomonic  projec- 
tion representing  the  forms  most  commonly  observed  on 
epidote.  A  crystal  repre- 
senting all  the  forms  pro- 
jected is  illustrated  in 
Figure  170.  The  plane 
of  projection  is  at  right 
angles  to  the  vertical  axis. 
The  distribution  of  the  pro- 
jection points  is  symmet- 
rical with  regard  to  one 
line  which  represents  the 
trace  of  the  plane  of  sym- 
metry. For  the  monoclinic 
system  the  centre  or  pole 

of  the  projection  is  not  a  projection  point  for  any  crystal 
face.  In  this  regard  the  monoclinic  projection  differs  from 
those  of  the  four  previous  systems.  There  are  numerous 
zone  lines,  some  right  and  left,  others  fore  and  aft.  These 
two  chief  series  of  zone  lines  intersect  at  right  angles.  Except 
the  line  of  symmetry,  which  usually  represents  a  prominent 
zone,  no  other  zone  line  passes  through  the  pole  of  projection. 
The  measurement  to  the  right  and  left  of  the  line  of  symmetry 
is  made  in  terms  of  paces  designated  q'0  and  all  the  distances 
to  the  right  or  left  of  that  line  are  simply  and  rationally  re- 


\                 °° 

0                   / 

Iff 
18 

J           « 

22                    1! 

1 

s? 

1                  22 

A 

0 

/     i 

o     N 

i 

sq 

0 

1 

\       0 

0        / 

E 

po 

^ 

1 

^qv 

—  \ 

8 
8 

/                                00 

0                   \ 

Fig.  169 


132  CRYSTALLOGRAPHY 

lated  to  one  another.  It  will  be  observed  that  a  fore  and  aft 
pace  may  be  employed  to  measure  conveniently  the  distance  be- 
tween any  pair  of  the  right  and  left  zone  lines.  Such  a  dis- 
tance is  p'0.  If  a  new  monoclinic  crystal  were  being  described 

the  pace  lengths  p'0  and  q'0  might  be 
arbitrarily  selected  from  the  proj  ection,, 
but  in  dealing  with  crystals  already 
described,  it  is  necessary  to  determine 
the  symbols  for  one  or  two  faces  so 
as  to  fix  the  pace  lengths  p'0  and  q'0 
before    proceeding    to    indicate    the 
Fig.  170  —  Epidote  show-  symbols  for  all  the  other  faces  or  to 
ing    the    orthopinacoid,    a,   graphically  determine  by  a  system  of 
(100);  the  negative  hemior-   ayera          the    yalues  for       ,     and      / 
thodome,  e,  (101);  the  posi-    ~.          .       .  .  . 

tivehemiorthodomes,  J,  (201)  Smce  m  thls  system  no  riSht  and  left 
and  r,  (101);  the  prism,  m,  zone  line  passes  through  the  polar 
(110);  the  prism,  u,  (210);  point  it  becomes  necessary  to  select 
the  clinodome,  o,  (Oil);  the  some  other  zone  line  from  which  the 
negative  hemipyramid,  d,  measurements  forward  and  backward 
(111);  four  positive  hemipy-  TT 

ramids,n,  (111);  x,  (112)  q,  are  to  be  made'  Usually  the  first 
(221)  and?/,  (211).  prominent  right  and  left  zone  line 

lying  in  front  of  the  polar  point  is  the 

line  of  reference  from  which  forward  and  backward  measure- 
ments are  made.  This  line  passes  through  the  projection  points 
for  the  following  faces:  (001),  (Oil)  and  (Oil)  and  at  infinity 
through  (OlO)  and  (010).  Having  selected  the  line  from 
which  fore  and  aft  measurements  are  to  be  made  and  the 
pace  lengths  p'0  and  q'0  the  symbols  may  then  be  assigned  to 
all  the  faces  and  the  polar  elements  p0  and  qo  determined. 
The  obliquity  between  the  a  and  c  axes  is  indicated  in  each 
projection  by  the  distance  between  the  pole  and  the  projec- 
tion for  the  face  (001).  This,  value  is  designated  e',  which  is 
-equal  to  the  tangent  of  p  for  (001).  The  values  p'0  and  q'0 
and  e'  as  measured  from  the  projection  are  expressed  in  terms 
of  the  height  of  plane  of  projection  h'0.  It  is  the  custom  of 
crystallographers  to  express  these  values  in  terms  of  the 
length  of  the  normal  of  (001)  as  measured  from  the  crystal 
centre  to  its  projection  point.  This  may  be  done  by  multiply- 
ing these  values  by  sin.  /3.  The  ratios  so  obtained  are  p0,  qo,  h0 
and  e. 


MONOCLINIC    SYSTEM  133 

f» 

qo  =  c.  sin.  /3;  p0  =  - 
a 

HEMIHEDRAL  CLASS 

It  is  possible  to  select  one-half  of  the  four  faces  of  the  hemi- 
pyramid  of  this  system  in  three  ways  as  illustrated  in  Figure 
165.     The  selection  of  the  faces  111 
and  111  or  111  and  111  would  give 
us  a  form   composed  of    two  faces 
parallel    to  one  another   and   sym- 
metrical about  the  centre  only.    This 
is  equivalent  to  the  holohedral  tri- 
clinic  class   and   will  be  considered 

in  connection  with  the  triclinic  sys-       „.     1>71      ~> 

Fig.  171  — Pyroxene  show- 
tern.  The  second  selection  of  one-  ing  hemihedral  developmeDt; 
half  the  four  planes  takes  place  in  forms  present  —  the  orthopin- 
such  a  way  that  the  plane  of  sym-  acoid,  a,  (100);  the  clinopina- 
metry  is  preserved,  the  centre  and  coid'  fe>  (°10^  the  Prism»  m> 

diad  axes  having  disappeared.     The    <110);  half °f  the  fa?es  of  ^e 

—  hemipyramids   o   and   u;    the 

extension  of  Cither  111  and   111  or    basal  pinacoid  c. 
of  111  and  111  gives  rise  to  this  type 

of  symmetry.  There  are  two  congruent  forms  from  each  of 
the  monoclinic  holohedra,  except  the  clinopinacoid.  Pyroxene 
which  belongs  to  this  class  is  represented  in  Figure  171. 

HEMIMORPHIC  CLASS 

The  forms  characteristic  of  this  class  may  be  derived  from 
various  holohedra  by  the  selection  of  those  parts  of  a  crystal 
form  on  one  side  of  the  plane  of  symmetry.  Here  the  diad 
axis  is  the  only  element  of  symmetry.  This  axis  is  hemi- 
morphic  since  the  faces  intersecting  one  end  of  this  axis  are 
not  necessarily  accompanied  by  similar  faces  at  the  opposite 
end.  Those  holohedral  forms  whose  faces  are  parallel  to  the 
b  axis  are  represented  by  forms  geometrically  identical  with 
those  of  the  holohedral  class.  On  the  other  hand  all  holohe- 
dral forms  which  intersect  the  b  axis  at  a  finite  distance  are 
here  represented  by  pairs  of  hemimorphic  forms  either  of 
which  may  occur  independently  of  the  other.  The  forms 
peculiar  to  this  class  are  as  follows: 


134 


CRYSTALLOGRAPHY 


1.  Negative  tetra-pyramids  —  right  (hkl)  and  left  (hkl), 

2.  Positive  tetra-pyramids  —  right  (hkl)  and  left  (hkl), 

3.  Hemi-prisms  —  right  (hkO)  and  left  (hkO),_ 

4.  Hemi-clinodomes  —  right  (Ohk)  and  leftjOhk), 

5.  Clinopinacoids  —  right  (010)  and  left  (OlO). 

Two  is  the  maximum  number  of  faces  belonging  to  a  crystal 
form  in  this  class,  while  for  the  hemi-clinopinacoid  the  form 
is  composed  of  a  single  face. 

Tartaric     acid,    C4H6O6,    quercite,    C6Hi2O5,    (Figure     172) 
and    cane    sugar    crystallize    in    the    hemimorphic    class.      If 
sodium  ammonium  racemate  be  allowed 
to  crystallize  from  a  solution  of  water, 
large   crystals  are  produced  similar  to 
those  of  tartaric  acid  which  are  repre- 
sented in  Figure  173.    It  will  be  observed 
that   the  crystals  here  represented  are 
similar  to  one  another  and  are  disting- 
Fig.  172  -Quercite  ex-    ujshed   by  the   right    and    left    handed 
hibitinghemimorphismon    arrangement  of  the  faces.     The  one  is 
the    axis    of    symmetry.  .  r    j.u         .1.1.  -re 

Forms  present:  the  prism,  the  mirror  imaSe  of  the  other'  If  a 
m,  (110);  the  basal  pina-  solution  be  prepared  exclusively  from 
coid,  c,  (001);  the  positive  crystals  of  the  one  class  it  is  found  to 
hemiorthodome,  p.  (101)  turn  the  piane  of  polarization  to  the 
rf  ht  hile  gimilar  solutionof  the  other 

' 
turns  it  in  the  opposite   direction.     If, 

however,  a  solution  be  prepared  from  equal  quantities  of 
substances  representing  the  two  types  the  resulting  solution  is 


and  the  right  hemiclino- 
dome,  e,  (Oil). 


Fig.  173  —  Tartaric  acid  showing  right  and  left  handed  enantiormorphous 
forms.  The  forms  present  are  —  the  orthopinacoid,  a,  (100) ;  the  basal 
pinacoid,  c,  (001);  the  unit  prism,  m,  (110);  the  negative  hemiorthodome, 
y,  (101);  the  positive  hemiorthodome,  p.  (101).  On  the  first  crystal  the  hemi- 
clinodome,  e,  (Oil),  appears  while  on  the  second  the  corresponding  form 
e',  (Oil),  is  present. 


MONOCLINIC   SYSTEM  135 

inactive  with  regard  to  polarized  light.  It  was  Pasteur 
who  first  demonstrated  the  relationship  between  geometrically 
right  handed  and  left  handed  crystals  and  dextro-rotary  and 
lavo-rotary  substances.  Racemic  acid  is  a  mixture  of  equal 
proportions  of  two  distinct  tartaric  acids  possessing  the  same 
chemical  composition  and  individually  giving  rise  to  crystals 
showing  this  right-handed  and  left-handed  character.  Lavo- 
rotary  and  dextro-rotary  organic  substances  are  known  as 
stereoisomers.  Substances  crystallizing  in  this  class,  being 
characterized  by  a  hemi-morphic  axis  of  symmetry,  show 
pyro-electricity  as  a  result  of  change  of  temperature. 


CHAPTER  XIII 

TRICLINIC   SYSTEM 

Only  two  classes  of  crystals  are  included  in  the  triclinic 
system.  One  of  these,  the  holohedral  triclinic  class,  is  char- 
acterized by  symmetry  about  a  centre.  The  other  class,  which 
is  hemihedral  triclinic,  is  marked  by  the  absence  of  all  sym- 
metry. In  the  latter  class  any  face  may  appear  without  the 
compulsory  appearance  at  the  same  time  of  any  other  face. 
The  crystal  form  consists  of  only  one  face. 

HOLOHEDRAL  CLASS 

Symmetry.  The  symmetrical  arrangment  of  faces  in  pairs 
about  a  centre  is  peculiar  to  the  holohedral  triclinic  class. 

The   various    crystal   forms   are 
^  composed  of  pairs  of  faces  so  that 

in  a  way  they  may  all  be  re- 
garded as  pinacoidal.  In  con- 
nection with  this  fact  it  may  be 
observed  that  this  class  is  some- 
times referred  to  as  triclinic 
Fig.  174— Albite— The  heavy  P^acoidal  in  contrast  to  the 
lines  indicate  the  vertical  zone  edges  class  without  symmetry,  which 
of  the  crystal  which  correspond  to  js  known  as  the  triclinic  asym- 
the  vertical  axis  c;  the  dotted  lines  metric  dags> 
similarly  correspond  to  the  fore  and  **.+•+*•  TT 

aft  axis  a  while  the  long  right  and  Standard  Orientation.  Very 
left  lines  show  the  direction  chosen  great  latitude  is  permitted  to 
for  the  right  and  left  axis,  6.  Forms  the  original  investigator  in  the 
present  — the  basal  pinacoid,  c,  case  of  triclinic  crystals.  As 
(001);thebrachypinacoid6  (010);  there  ^  nQ  laneg  Qr  axeg  of 
the  right  hemipnsm,  m,  (110);  the  A,  .  ,  n  ,t 

lefthemiprism,  M,  (Ho);  the  tetra-  symmetry  there  is  no  very  definite 
pyramid,  o,  (111)  and  the  hemimac-  regulation  as  to  the  orientation 
rodome,  x,  (403).  of  such  crystals.  In  triclinic 


TRICLINIC    SYSTEM  137 

crystals  of  both  classes  there  is  a  complete  absence  of  rectangu- 
larity,  no  plane  or  edge  being  at  right  angles  to  any  other 
plane  or  edge.  In  selecting  the  standard  orientation  of  a  new 
triclinic  substance  it  is  usual  to  pick  out  three  sets  of  zone 
edges  intersecting  one  another  approximately  at  right  angles, 
and  to  so  orient  the  crystal  that  these  three  edge  directions 
are  approximately  fore  and  aft,  right  and  left  and  vertical. 
Figure  174  exhibits  a  crystal  of  albite  with  the  three  edge 
directions  shown  in  lines  of  different  type. 

Axes  of  Reference.  The  faces  of  triclinic  forms  are  referred 
to  three  inequivalent  axes  parallel  to  the  three  series  of  zone 
edges  selected  as  a  basis  for  orientation. 
As  indicated  in  Figure  175  the  crystal  is 
usually  so  oriented  that  the  obtuse  angles 
formed  by  the  three  axes  are  all  enclosed 
by  the  positive  ends  of  the  axes.  The 
terminology  in  regard  to  the  axes  of  refer- 
ence for  the  triclinic  system  is  similar  to 
that  employed  for  the  rhombic  system. 
The  axes  a,  b  and  c  are  designated  brachy,  F.  "c 

macro    and    vertical    axes,    respectively. 

Since  the  angles  between  the  various  pairs  of  axes  are  always 
oblique  in  this  system  it  is  convenient  to  indicate  the  magni- 
tude of  those  angles  by  means  of  the  three  letters  a,  0  and  7. 
a  is  the  angle  between  the  b  and  c  axes, 
|8  is  the  angle  between  the  a  and  c  axes, 
7  is  the  angle  between  the  a  and  b  axes. 

The  letter  0  was  used  in  the  monoclinic  system  to  indicate 
the  value  of  the  angle  between  the  a  and  c  axes. 

Classification  of  All  Possible  Planes  Intersecting  Triclinic 
Axes  of  Reference. 

(a)  The  plane  intersects  only  one  axis: 

(1)  a  axis  cut  —  macropinacoid, 

(2)  b  axis  cut  —  brachypinacoid, 

(3)  c  axis  cut  —  basal  pinacoid 

(6)  The  plane  intersects  two  axes: 

(1)  a  axis  not  cut  —  two  possibilities  — 

1st.     Plane  opposite  obtuse  angle  a  —  right  hemi-brachydome, 
2nd.     Plane  opposite  acute  angle  a  —  left   hemi-brachydome, 


138  CRYSTALLOGRAPHY 

(2)  b  axis  not  cut  —  two  possibilities  — 

1st.     Plane  opposite  obtuse  angle  ft  —  upper  hemi-macrodome, 
2nd.     Plane  opposite  acute  angle  ft  —  lower  hemi-macrodome, 

(3)  c  axis  not  cut  —  two  possibilities  — 

1st.     Plane  opposite  obtuse  angle  7  —  right  hemi-prism, 
2nd.     Plane  opposite  acute  angle  7  —  left  hemi-prism, 

(c)  The  plane  intersects  three  axes  —  four  possibilities: 

1st.  Right  upper  tetra-pyramid, 

2nd.  Left  upper  tetra-pyramid, 

3rd.  Right  lower  tetra-pyramid, 

4th.  Left  lower  tetra-pyramid. 

There  are,  therefore,  in  the  triclinic  system,  owing  to  the 
low  degree  of  symmetry,  thirteen  distinct  ways  in  which 
planes  may  intersect  the  axes  of  reference. 

Tetra-pyramids.  The  forms  belonging  to  each  of  these 
four  types  are  composed  of  pairs  of  planes  parallel  to  one 
another  occurring  on  opposite  sides  of  a  crystal.  The  rhombic 
pyramid,  which  is  bounded  by  eight  planes,  corresponds  to  the 
eight  planes  on  the  positive  and  negative  hemi-pyramids  of 
the  monoclinic  system  and  to  the  eight 
planes  of  the  four  tetra-pyramids  of  the 
triclinic  system.  These  four  forms  when 
occurring  together  enclose  space  giving  rise 
to  a  combination  as  represented  in  Figure 
176.  When  the  original  investigator  has 
selected  a  unit  tetra-pyramid  he  inciden- 
tally settles  the  ratios  of  the  lengths  of  the 
intercepts  on  the  three  axes  of  reference,  and 
Fig.  176  after  this  arbitrary  decision  has  been  reached 

the  six  crystallographic  constants  of  the  tri- 
clinic system  are  fixed.     These  characteristic  constants  for  the 
triclinic  mineral  albite,  NaAlSi3O8,  are  as  follows: 
a:b:c:  :  .6335:1:  .5577, 
a  =  94°  3':  0  =  116°  29':  7  =  88°  9'. 

The   symbols  for  the   unit  forms,   as   well   as  the   general 
symbol  for  each  of  the  four  tetra-pyramids,  are  as  follows: 
Right  upper  tetra-pyramid  —  (HI)  and  (hkl), 
Left  upper  tetra-pyramid       -  (ill)  and  (hkl), 
Right  lower  tetra-pyramid  —  (111)  and  (hkl), 
Left  lower  tetra-pyramid       -  (111)  and  (hkl). 


TRICLINIC    SYSTEM 


139 


Hemi-macrodomes.  There  are  in  this  system  two  hemi- 
macrodomes,  each  composed  of  a  pair  of  faces  corresponding 
to  the  pair  of  hemi-orthodomes  characteristic  of  the  last 
system.  Symbols: 

Upper  hemi-macrodome,  unit  form  (101), 

general  symbol  (hOk), 
Lower  hemi-macrodome,  unit  form  (101), 

general  symbol  (hOk). 

Hemi-brachydomes.  These  forms  are  similar  in  their 
occurrence  to  the  pair  of  forms  just  described.  Sym- 
bols: 

Right  hemi-brachydome  —  unit  form  (Oil)  — 

general  symbol  (Ohk), 
Left  hemi-brachydome  —  unit  form  (Oil)  - 

general  symbol  (Ohk). 

Hemi-prisms.  The  relation  of  these  prisms  to  one  another 
is  the  same  as  that  which  maintains  between  the  pair  of 
brachydomes.  Symbols: 

Right  hemi-prism  —  unit  form  (110)  — 

general  symbol  (hkO), 
Left  hemi-prism  —  unit  form  (110)  — 

general  symbol  (hkO). 

Pinacoids.     The  three  pinacoids  bear  the  same  relationship 
to   one   another  as   do   the  three   pinacoids  in  the   last  two 
systems.     Their  mode  of  intersection  when 
in  combination  with  one  another  is  shown 
in  Figure  177.      These  three  forms  are  in- 
variable in  contrast  to  those  types  already 
considered,  where,  in  addition  to  the  unit 
form,  a  large  number  of  other  forms  are 
possible,  such  as  have  been  indicated  by 
the  use  of  the. general  symbols.     Symbols  : 
Macropinacoid  (100), 
Brachypinacoid  (010), 
Basal  pinacoid    (001). 
Combinations.     In   this   class   where   each   crystal   form   is 
made  up  of  only  two  faces  there  are  no  closed  forms,  so  that 
every    triclinic    crystal    must   present   two    or    more    distinct 
forms.     Figures  178,  174,  179  and  180  show  the  combinations 
of   triclinic  forms    observed   on    crystals   of   anorthite,  albite, 


-t— 


Fig.  177 


140 


CRYSTALLOGRAPHY 


Fig.  178 — Anorthite  exhibiting  the 
basal  pinacoid,  c,  (001);  the  brachy - 
pinacoid  6,  (010);  the  right  hemi- 
prism,  ra,  (110);  the  left  hemiprism, 
M,  (110);  the  right  hemiprism,  /, 
(130);  the  left  hemiprism,  z,  (130); 
the  right  hemibrachy domes,  e,  (021) 
and,  r,  (061);  the  left  hemibrachy- 
dome,  n,  (021);  and  the  tetrapyra- 
mids,  a,  m,  p,  o,  and  v. 


Fig.  179  — Copper  sulphate,  CuS04 
+  5H2O,  showing  the  basal  pinacoid, 
c,  (001);  the  brachypinacoid,  6, 
(010);  the  macropinacoid,  a,  (100); 
the  left  hemiprism  M,  (lIO);  the  left 
hemiprism,  /,  (120);  the  right  hemi- 
prism, m,  (110)  and  the  right  front 
upper  tetrapyramid,  s,  (111). 


copper  sulphate,   CuSO4  +  5  H2O,  and  calcium  hyposulphite, 

CaS203  +  6  H2O,  respectively. 

Limiting  Forms.  In  this  system  the 
pyramid  is  the  most  general  form  and 
by  varying  the  intercept  on  the  verti- 
cal axis  a  series  of  pyramids  is  ob- 


tained which  finds  its  limits  in  the 
basal  pinacoid  and  prism.  Similarly 
by  varying  the  intercept  on  the  macro- 
Fig.  180-  Calcium  hypo-  axis  various  pyramids  are  obtained 
sulphite  showing  the  basal  limited  by  the  brachypinacoid  and 
pinacoid,  c,  (001) ;  the  brachy-  macrodome.  In  the  same  way  by  vary- 
pinacoid,  6,  (010);  the  right  ing  the  intercept  on  the  brachy  axis 
hemiprism,  m,  (110);  the  left  we  obtain  a  series  of  pyramids  whose 
hemiprism,  M,  (110)  and  the  ,.  ..  ,,  .  .  ,  ,  ,, 

right    hemibrachydome,    o,   hmits  are  the  macropinacoid  and  the 
(Oil).  brachydome.      In  this  system,  as  in 

the  last,  variations  of  the  domes  and 
prisms  reach  their  limit  in  pairs  of  pinacoids. 

Projections.  Figure  181  represents  a  gnomonic  projection 
for  the  crystal  of  anorthite,  shown  in  Figure  178.  It  may  be 
observed  that  in  this  projection  the  points  occur  on  zone 
lines  but  there  is  no  symmetrical  arrangement  about  any  line 
or  point.  This  corresponds  to  the  known  symmetry  of  the 
triclinic  holohedral  class.  The  zone  lines  do  not  intersect 


TRICLINIC    SYSTEM 


141 


at  right  angles,  which  is  in  accord  with  the  general  obliquity 
characteristic  of  the  system.  The  pole  of  the  projection  is 
not  the  projection  point 
for  any  face,  neither  does 
any  zone  line  pass  through 
the  pole.  The  fore  and 
aft  pace  length  p'0  is  indi- 
cated by  half  the  distance 
between  the  points  for  11 
and  11,  while  the  right 
and  left  pace  q'0  corres- 
ponds to  half  the  distance 
between  11  and  11,  and 
h'o  equals  radius  of  the 
ground  circle.  From  these  lengths,  which  may  be  measured  on 
a  gnomonic  projection,  the  polar  elements  p0:  q0:  r0  may  be 

derived.  Owing  to  the  obliquity 
of  all  the  axes  of  reference  crys- 
tallographic  calculations  in  con- 
nection with  the  triclinic  system 
are  somewhat  more  difficult  than 
those  of  the  other  systems.1 


\OODO                        65  oo/           1 

V 

2i 
/ 

1 

-v 

1      -I 

Pol 

} 

02\ 

^ 

7    01                              00 

/ 

1 

0 

V 

/OOOO                                   ODOO\ 

Fig.  181 


HEMIHEDRAL  CLASS 

Fig.  182  -  Acid  strontium  tart-  In  this  class  are  comprised  those 

rate.     HemihedraJ  triclinic,  exhi-  ^  whoge  distribution  of  faceg 

biting  the  half  basal  pmacoid,  c,  .    * 

(001);  the  half  brachy pmacoid,  b,  1S  such   as  to   indicate  the  entire 

(010);  the  half  macropinacoid,  a,  absence  of  symmetry.     Any  face 

(100);   the  pyramid,  u,  (122)  and  which  is  in  accord  with  the  ration- 

the  lower  macrodome,  /,    (101).  ality  of  axial  parameters  may  oc- 

Each  of  these  crystal  forms  con-  cur  unaccompanied  by             other 

sists  of  a  single  face.  /         * 

face.      The   crystal  form  is   here 

composed  of  a  single  plane.  As  a  matter  of  fact  on  crystals 
of  the  few  substances  belonging  to  this  class  faces  frequently 
occur  in  pairs  but  their  opposite  and  parallel  faces  are  physically 
distinct  forms  and  their  simultaneous  occurrence  is  more  or 


1  For  an  example  of  calculations  for  this  system  see  Goldschmidt  and 
Borgstrom,  "  Zeitschrift  fiir  Krystallographie."     Band  XL1. 


142  CRYSTALLOGRAPHY 

less  accidental.  The  substances  belonging  to  this  class  are 
all  chemically  complex  so  that  they  stand  in  a  sharp  contrast 
to  most  of  the  substances  crystallizing  in  the  cubic  system 
where  .the  degree  of  symmetry  is  very  high.  They  are  all 
products  of  the  chemical  laboratory.  Figure  182  indicates  the 
form  of  crystals  of  strontium  bitartrate,  Sr(C4H406H)2  +4H2O. 


CHAPTER   XIV 


CRYSTAL   AGGREGATES 

PARALLEL  GROWTHS 

There  is  a  marked  tendency  for  several  crystals  whose 
corresponding  directions  are  all  parallel  to  one  another  to 
grow  together  into  one  mass,  while 
the  individuality  of  the  various  crys- 
tals is  shown  on  certain  portions  of 
the  surface  by  the  occurrence  of  the 
faces  characteristic  of  each.  This  is 
often  the  case  with  the  mineral 
quartz  (Figure  183).  This  method 
of  aggregation  is  described  as  a  par- 
allel growth. 

Crystals  of  different  substances  be- 
longing to  the  same  crystal  system 
sometimes  are  grown  together  in  such 


Fig.  183  —  Quartz  showing 
parallel  growth. 


a  way  that  the  directions  of  the  axes  of  reference  are  common  to 
both,  certain  edges  and  planes  of  the  one  crystal  being  parallel 
to  the  corresponding  edges  and  planes  of  the  other.  In  this 
case,  although  the  substances  are  differ- 
ent from  one  another,  a  complete  paral- 
lelism in  orientation  maintains.  This  is 
shown  with  remarkable  perfection  in  Fig- 
ure 184,  exhibiting  a  crystal  of  zircon  en- 
closed in  a  crystal  of  xenotime.  Both 
these  minerals  are  of  the  tetragonal  sys- 
tem and  they  are  so  intergrown  that 
the  corresponding  tetragonal  directions 
of  the  two  substances  are  parallel. 

A  parallelism  less  perfect  is  sometimes  exhibited  by  sub- 
stances belonging  to  different  crystal  systems  where  certain 
planes  and  edges  are  common  to  the  two  crystals.  This  has 


Fig.  184  —  Parallel  in- 
tergrowth  of  zircon  and 
xenotime. 


144  CRYSTALLOGRAPHY 

been  frequently  observed  on  crystals  of  staurolite  and  kyanite 
from  the  St.  Gothard  region  in  Switzerland.  The  former  of 
these  minerals  is  rhombic,  the  latter  triclinic. 


IRREGULAR  AGGREGATES 

In  rocks  containing  several  prominent  minerals  there  is 
usually  no  perceptible  regularity  in  the  arrangement  of  the 
crystals  with  regard  to  one  another.  Where  the  crystal 
individuals  have  grown  and  developed  until  they  come  into 
contact  with  other  individuals  there  is  a  general  absence  of 
crystal  faces,  the  boundaries  between  the  individuals  being 
quite  irregular.  This  lack  of  direction  in  the  orientation 
of  the  crystal  units  is  exceedingly  common  in  rocks  and  is 
sometimes  referred  to  as  irregular  aggregation. 

FIBROUS  RADIATING  AGGREGATES 

Those  minerals  characterized  by  acicular  crystals  are  often 
found  in  more  or  less  spherical  or  hemispherical  masses  built 
up  of  needles  radiating  from  a  centre.  This  method  of  aggre- 
gation is  very  characteristic  for  natrolite,  wavellite  and 
hematite. 

TWINNING 

On  simple  crystals  such  as  have  been  described  in  connec- 
tion with  the  various  systems  there  is  a  general  absence  of 
re-entrant  angles  and  it  has  been  assumed  that  all  the  particles 
of  which  a  crystal  is  built  up  are  absolutely  parallel  to  one 
another.  However,  we  frequently  find  crystals  characterized 
by  re-entrant  angles  which  may  show  cleavage  in  different 
directions  in  different  parts  of  what  is  apparently  the  same 
crystal.  Such  crystals  are  said  to  be  twins  and  are  made 
up  of  two  or  more  crystal  individuals  grown  together  in 
some  regular  way.  A  very  simple  case  of  this  kind  is 
exhibited  by  the  swallow-tailed  twin  of  gypsum  (Figure 
185).  If  this  twin  of  gypsum  be  divided  into  two  parts 
by  the  plane  marked  by  the  dotted  lines,  and  one  of  the 
parts  be  revolved  180°  about  a  line  normal  to  the  dividing 
plane,  the  two  parts  in  their  new  position  will  appear 


CRYSTAL   AGGREGATES 


145 


to  represent  only  an  ordinary  crystal  of  this  substance. 
To  explain  forms  of  this  type  an  hypothesis  has  been  sug- 
gested according  to  which  early  during  the  growth  of  the 


Fig.  185  —  Showing  the  derivation  of  the  swallow-tailed  twin  of  gypsum 
from  the  simple  crystal  individual  by  dividing  the  latter  by  a  plane  parallel 
to  the  orthopinacoid  and  then  revolving  one  half  about  a  line  normal  to  this 
plane. 

crystal  two  crystal  particles  as  represented  in  Figure  186 
were  brought  together  in  such  a  way  as  to  secure  parallelism 
along  two  directions,  while  the  third  directions  are  not 
parallel.  From  this  partial  parallelism  of  the  two  crystal 
building  particles,  the  crystal  in  its  complex  form  has  been 
produced  —  each  particle  attracting  to  itself  and  orienting 
in  its  own  fashion  other  particles  of  the  same  substance.  As 


Fig.  186  —  Diagram  showing  partial  parallelism  between  two  particles. 

the  two  original  particles  were  symmetrical  with  regard  to 
the  plane  between  them,  so  are  the  resultant  half  crystals 
symmetrical  about  this  same  plane.  The  plane  about  which 


146  CRYSTALLOGRAPHY 

the  two  halves  of  the  twin  crystals  are  symmetrically  arranged 
is  always  a  possible  crystal  face  of  the  substance  in  question, 
that  is,  the  direction  of  the  twinning  plane,  as  it  is  called,  is 
such  as  to  intersect  the  axes  of  reference  in  accordance  with 
the  law  of  rationality  of  axial  parameters.  The  twinning 
plane  obviously  can  never  have  the  direction  of  a  plane  of 
symmetry  on  the  simple  crystal.  Twin  crystals,  such  as 
that  of  gypsum,  present  a  higher  degree  of  symmetry  than 
the  individual  crystals  from  which  they  are  derived.  The 
swallow-tailed  twin  of  gypsum  (Figure  185)  is  characterized 
by  the  presence  of  two  planes  of  symmetry  and  one  diad 
axis  of  symmetry,  simulating  in  this  way  a  higher  grade  of 
symmetry.  This  is  a  general  effect  of  twinning. 

The  normal  to  the  twinning  plane  about  which  the  half 
of  the  twin  may  be  revolved  so  as  to  become  parallel  to  the 
other  is  known  as  the  twinning  axis.  Twin  crystals  may  be 
supposed  to  be  produced  by  placing  two  'crystal  individuals 
of  the  same  substance  parallel  to  one  another  and  revolving 
one  of  them  180°  about  its  twinning  axis.  The  plane  along 
which  these  two  individuals  in  their  new  orientations  are 
brought  into  contact  with  one  another  is  known  as  the  com- 
position face.  Usually  the  composition  face  is  identical  with 
the  twinning  plane  but  in  certain  types  of  twinning  for 

orthoclase  the  composition   face  and 
twinning  plane  are  not  identical. 

The  statement  of  the  directions  of 
the  twinning   plane  composition   face 
and  twinning   axis  for  any  particular 
substance  is   known  as   the  twinning 
law  for  that  substance.     Certain  sub- 
Fig.  187— Fluorite  exhibit-  stances  such    as  orthoclase   twin  ac- 
ing  interpenetration  twinning    cording  to  several  distinct  laws. 

of  two  cubes  with  the  twin-       TWO  Chief  Types  of  Twinning.     In 

ning  axis  normal  to  the  face    the    commonest    form    of   twinning  il- 
of  the  octahedron.  ,  .._-..  , 

lustrated  by  Figure  185  the  two  indi- 
viduals are  merely  brought  into  contact  with  one  another. 
There  is  a  surface  more  or  less  regular  which  divides  the 
twin  crystal  into  two  such  parts  that  all  the  particles  on  each 
side  of  that  surface  have  a  common  orientation.  This  mode 
of  aggregation  gives  rise  to  contact  twins.  Another  type  of 


CRYSTAL  AGGREGATES 


147 


twinning  is  illustrated  by  Figure  187,  representing  two  cubes  of 
fluorite,  so  intergrown  as  to  have  a  common  centre.  In 
this  case  there  is  no  surface  dividing  particles  of  the  two  orien- 
tations from  one  another.  Throughout  the  central  part  of  the 
twin  crystal  the  portions  representing  the  different  orienta- 
tions are  irregularly  intermingled.  This  type  of  intergrowth,  in 
which  the  general  outline  of  the  one  crystal  includes  a  large 
proportion  of  the  substances  contained  within  the  general 


Fig.  188  —  Interpenetration  twin 
of  staurolite  —  twinning  plane  the 
brachydome  (032). 


Fig.  189  —  Albite  showing  repeated 
contact  twinning  according  to  the 
albite  law. 


outline  of  the  other,   gives  rise  to  penetration  twins.     Figure 
188  indicates  this  type  for  staurolite. 

Repeated  Twinning.  The  process  which  gives  rise  to  two 
distinct  individuals  showing  partial  parallelism  is  often 
repeated  as  in  the  case  of  albite.  Figure  189  represents  the 
common  twinning  of  albite,  where  the  two  individuals  are 
placed  side  by  side,  with  a  brachypinacoid  as  the  composi- 
tion face.  In  albite  this  process  is  repeated  indefinitely  so 
that  the  twin  crystal  is  composed  of  a  large  number  of  thin 
plates  placed  side  by  side  in  such  a  way  that  the  1st,  3rd,  5th, 
7th,  etc.,  are  absolutely  parallel  to  one  another,  while  the  2nd, 
4th,  6th  and  8th  have  a  similar  relationship  to  one  another. 
This  is  a  case  of  repeated  twinning  in  which  the  alternate 
individuals  are  parallel.  In  nature  there  is  a  marked  tendency 
for  the  individual  in  such  repeated  twins  to  be  shortened  in 
a  direction  at  right  angles  to  the  composition  face.  Each 
of  these  twin  plates  is  a  lamella  and  this  type  of  repeated 
twinning  is  known  as  lamellar  or  polysynthetic  twinning. 
Aragonite  frequently  twins  in  this  fashion. 


148 


CRYSTALLOGRAPHY 


A  second  type  of  repeated  twinning  results  from  the  twin- 
ning of  the  successive  individuals  according  to  the  same 
twinning  law  except  that  the  twinning  plane  for  any  pair  of 
individuals  is  not  the  same  as  that  for  any  preceding  pair. 
The  twinning  plane  is  another  face  of  the  same  crystal  form. 


Fig.    190 
ning. 


Cyclic    contact   twin- 


Fig.  191  —  Cyclic  contact  twinning 
as  shown  on  rutile  —  twinning  plane 
a  face  of  the  pyramid  of  the  second 
order  (101). 


Here  the  alternate  individuals  are  not  parallel  to  one  another. 
Figure  190  represents  diagrammatically  a  cross-section  through 
such  a  series  of  twins.  This  type  of  repeated  twinning  tends 
to  arrange  the  successive  members  in  a  circular  fashion  and 
is  known  as  cyclic  twinning.  Figure  191  represents  this  type 
of  twinning  as  applied  to  the  mineral  rutile.  It  is  proposed 
in  the  following  pages  to  consider  the  chief  twinning  laws 
of  the  different  crystal  systems  and  to  illustrate  these  laws 
by  reference  to  mineral  examples. 


COMMON  TWINNING  LAWS 

CUBIC  SYSTEM 

Spinel,  MgAl2O4.  The  commonest  twinning  law  of  the 
cubic  system  is  that  frequently  observed  on  spinel,  magnetite, 
native  gold  and  native  copper.  This  is  represented  in  Figure 
192.  By  the  spinel  law  we  obtain  contact  twins  whose  twin- 
ning plane  and  composition  face  is  (111). 

Fluorite,  CaF2.  In  fluorite  (Figure  187)  the  twinning 
plane  is  also  (111)  the  same  as  for  spinel,  but  this  mineral 
gives  rise  to  penetration  twins.  If  one  of  the  individuals 
shown  be  revolved  180°  about  the  triad  axis  it  will  then 


CRYSTAL  AGGREGATES 


149 


occupy  such  a  position  that  all  its  edges  and  faces  are  parallel 
to  the  corresponding  edges  and  faces  of  the  other  individual. 

Pyrite,  FeS2.  The  very  beauti- 
ful twin  of  pyrite  in  Figure  193 
shows  penetration  twinning  with 
(110)  as  the  twinning  plane.  This 
peculiar  twin  is  known  as  the  iron 
cross.  It  should  be  noted  that  as 
a  result  of  twinning  the  pyrite  has 
attained  a  symmetry  much  higher 
than  that  characteristic  for  the  indi- 
vidual crystal.  Twinning  which  gives 
to  hemihedral  crystals  the  symmetry  characteristic  of  the 
holohedral  class  of  the  system  to  which  they  belong  is  known 


Fig.  192  —  Contact  twin- 
ning of  spinel  —  twinning 
plane  the  octahedron  (111). 


Fig.  193  —  The  Iron  Cross  twin  of  pyrite  —  twinning  plane  (110). 

as  supplementary  twinning.     The  iron  cross  twin  of  pyrite  is 
a  good  example  of  this  type. 


TETRAGONAL  SYSTEM 

Cassiterite,  Sn02.  This  tetragonal  mineral  very  seldom 
occurs  in  single  crystals.  The  twinning  of  cassiterite  may 
be  described  as  follows:  contact  twins  with  the  pyramid  of 
the  second  order  (101)  as  twinning  plane  and  composition 
face  (Figure  194).  Since  the  two  parts  of  the  crystal  are 
bent  towards  one  another  in  a  way  suggesting  a  knee  the 
cassiterite  twin  is  sometimes  spoken  of  as  a  geniculate  twin. 
Rutile,  Ti02,  is  frequently  twinned  according  to  the  same 
law  (Figure  191)  while  in  the  closely  related  mineral  zircon, 
ZrSi04,  twinning  according  to  this  law  is  quite  rare.  The 
crystals  of  the  mineral  rutile  commonly  twin  with  the  pyra- 
mid of  the  second  order  as  twinning  plane  and  composition 


150 


CRYSTALLOGRAPHY 


face,  but  in  such  a  fashion  that  while  the  twinning  is  repeated 
the  alternate  individuals  are  not  parallel.  For  the  second 
and  third  individuals  the  twinning  plane  while  still  a  plane 


Fig.  194  —  Genicu- 
late  twinning  of  cas- 
siterite  —  twinning 
plane  (101). 


Fig.  195  —  Cyclic 
twinning  of  r u t i  1  e . 
The  twinning  planes 
are  faces  of  the  pyra- 
mid (101). 


Fig.  196— Complete 
cycle  resulting  from  twin- 
ning shown  in  Figure  191. 


of  the  pyramid  of  the  second  order  is  not  the  same  plane  as 
that  which  served  as  twinning  plane  for  the  first  and  second 
individuals.  Figure  195  shows  a  repeated  twin  of  this  type 
composed  of  four  individuals.  If  the  process  of  cyclical 
twinning  be  continued  an  aggregate  such  as  Figure  196  may 
result. 

HEXAGONAL  SYSTEM 

Calcite,  CaCO3.      Calcite    exhibits    polysynthetic    twinning 
in  which  the  twinning  plane  and  composition  face  are  parallel 


Fig.  197  —  Calcite  showing  in  its 
simplest  form  twinning  with  the 
rhombohedron  of  the  first  order 
«(0ll2)  as  twinning  plane. 


Fig.  198 —  Contact  twinning  in  cal- 
cite  with  the  basal  pinacoid  (0001) 
as  twinning  plane. 


CRYSTAL  AGGREGATES 


151 


to  the  negative  rhombohedron  —  /c(0112).  Figure  197  illus- 
trates the  relative  positions  of  the  first  pair  in  twinning 
according  to  this  law. 

Another  method  of  twinning  exhibited  by  this  mineral  is 
shown  in  Figure  198.  For  this  form  the  twinning  plane  and 
composition  face  is  the  basal  pinacoid  (0001)  which  for  cal- 
cite  is  not  a  plane  of  symmetry. 

The  butterfly  twin  of  calcite  results  from  twinning  on  a 
steep,  negative  rhombohedron  —  *(0221). 

Quartz,  Si02.  Tetartohedral  crystals  of  quartz  frequently 
give  rise  to  penetration  twins  —  twinning  plane,  the  basal 


Fig.  199  —  Types  resulting  from  twinning  of  tetartohedral  crystals  of  quartz 
—  the  first  by  the  interpenetration  of  two  right  handed  individuals  with  (0001) 
as  twinning  plane  —  the  second  by  the  interpenetration  of  right  and  left  hand 
individuals  with  the  same  twinning  plane. 

pinacoid  (0001).  The  first  picture  in  Figure  199  represents 
the  twinning  of  two  right-handed  quartz  crystals.  Twinning 
of  crystals  of  opposite  types  give  rise  to  the  twin  illustrated 
in  the  second  picture  (Figure  199).  The  first  type  of  twin- 
ning in  the  case  of  quartz  is  by  the  Dauphine  law,  while  the 
second  is  according  to  the  Brazilian  law.  By  means  of  phys- 
ical tests  it  is  sometimes  possible  to  select  the  faces  or  parts 
of  faces  belonging  to  each  of  the  twinned  individuals. 


RHOMBIC  SYSTEM 

Aragonite,  CaCO3.  For  this  mineral  the  common  twin- 
ning plane  is  the  unit  prism  (110).  It  is  usually  repeated  and 
may  be  polysynthetic  or  cyclic  (Figure  190).  Frequently 
three  individuals  with  the  orientation  of  cyclic  twins  inter- 


152 


CRYSTALLOGRAPHY 


penetrate  so  as  to  give  rise  to  pseudo-hexagonal  forms.  This 
approximation  as  a  result  of  twinning  of  crystals  of  a  lower 
degree  of  symmetry  to  forms  of  a  higher  degree  of  symmetry 
is  known  as  mimicry.  In  this  case  the 
pseudo-hexagonal  form  is  due  to  the 
fact  that  the  prism  angle  for  aragonite 
is  nearly  60°.  The  mineral  cordierite 
produces  interpenetrating  .twins  similar 
to  those  of  aragonite. 

Staurolite,  Al2SiO5.  The  twinning 
plane  for  the  common  form  of  stauro- 
lite  is  the  brachydome  (032).  The  indi- 
viduals interpenetrate  with  the  resulting 
form  shown  in  Figure  188.  A  second 
twinning  law  .for  this  mineral  gives  rise 

to  an  oblique  cross  (Figure  200).     The  twinning  plane  for  this 
second  type  is  the  pyramid  (232). 


Fig.  200  —  Interpene- 
tration  twin  of  stauro- 
lite — twinning  plane  the 
pyramid  (232). 


MONOCLINIC  SYSTEM 

Gypsum,  CaSO4  +  2  H20.  The  swallow-tailed  twin  of 
gypsum  (Figure  185)  results  from  contact  twinning  when  the 
orthopinacoid  (100)  serves  as  twin- 
ning plane  and  composition  face. 
Many  monoclinic  minerals  twin  in 
this  fashion,  among  which  may  be 
mentioned  augite  (Figure  201),  horn- 
blende, malachite  and  epidote. 

Orthoclase,  KAlSiaOs.  This  mono- 
clinic  mineral  twins  according  to 
several  laws  which  may  be  stated  as 
follows,  in  order  of  their  importance: 


Fig.  201 — Augite  twinned 
on  the  orthopinacoid  (100). 


1.  Karlsbad  law.  —  twinning  plane  the  orthopinacoid  (100);   composition 
face,    the    clinopinacoid    (010),    partially    penetrating.      This    is    an 
exceedingly   common   and   characteristic  form   of  twinning  for  this 
mineral  (Figure  202  a  and  b). 

2.  Baveno  law.  —  twinning  plane  and  composition  face,  the  clinodome 
(021)   (Figure  202  d).     Crystals  of  orthoclase  showing  this  type  of 
twinning  are  usually  elongated  in  the  direction  of  the  clino  axis. 

3.  Mannebach   law.  —  twinning  plane    and   composition  face  the  basal 
pinacoid  (001)  (Figure  202  c). 


CRYSTAL  AGGREGATES 


153 


Fig.  202  —  Orthoclase  twinned  by  the  three  common  laws  for  this  species. 

Mica.  The  minerals  of  the  mica  group  twin  according  to 
the  mica  law,  in  which  the  twinning  plane  belongs  to  the 
zone  of  (110)  and  (001)  and  is  at  right  angles  to  the  basal 
pinacoid  (001),  while  the  composition 
face  is  (001).  The  individuals  are  placed 
on  top  of  one  another  as  shown  in 
Figure  203.  This  rather  complex  law 
may  seem  to  stand  in  conflict  with  the 
general  statement  according  to  which 
the  twinning  plane  is  always  a  possible 


7  — 

V 

-I 

Fig.  203  — Mica  con- 


.    ,    „  tact   twin   by   the   mica 

crystal  face.     In  this  law  the  direction    law 

of  the  twinning  plane   can  be  stated  in 

terms  of  possible  crystal  faces  but  the  twinning  plane  itself 

is  not  a  possible  crystal  face  in  the  monoclinic  system. 


TRI  CLINIC  SYSTEM 

Albite  (NaAlSi3O8).  The  mineral  albite  is 
one  of  the  feldspars  and  is  known  to  twin 
according  to  the  three  laws  previously  men- 
tioned for  orthoclase  but  it  most  frequently 
twins  according  to  the  albite  law  which  may 
be  stated  as  follows: 

Polysynthetic  twinning  with  twinning  plane 
and    composition    face    the    brachypinacoid 
Fig.  204  — Simple    (010). 

twin  according  to  the  This  type  of  twinning  can  be  readily  ob- 
albite  law  twin-  served  on  crystals  of  albite  and  also  on  the 
ning  plane  the  bra-  plagioclases  generally.  It  gives  rise  to  par- 
chypmacoid  (010).  ,,,....  Af  , 

allel  stnations  on  the  best  cleavage,  or  on 

the  natural  basal  pinacoid.     Figure  204  represents  two  indi- 


154 


CRYSTALLOGRAPHY 


viduals  twinned  in  this  way,  while  the  complex  polysynthetic 
aggregate  is  shown  in  Figure  189. 

In  the  previous  pages  only  a  few  of  the  commoner  types 
of  twinning  have  been  referred  to.  Twinning  according  to 
many  other  laws  has  been  observed  and  will  be  found  on 
record  as  a  part  of  the  regular  description  of  the  various 
species. 

General  Effect  of  Twinning.  It  has  already  been  observed 
that  the  effect  of  twinning  is  to  increase  the  symmetry  so 
that  certain  crystals  of  low  symmetry  as  a  result  of  complex 
twinning  simulate  a  very  high  degree  of  symmetry.  This 
is  particularly  well  shown  in  the  case  of  the  monoclinic 


Fig.  205  —  Mimicry  of  higher  forms  exhibited  by  the  monoclinic  mineral 
phillipsite  as  a  result  of  twinning. 

mineral  phillipsite.  Figure  205a  represents  a  common  form 
for  phillipsite  with  twinning  on  the  basal  pinacoid  (001). 
The  two  individuals  interpenetrate  so  that  the  diagonally 
opposite  quarters  seen  on  the  clinopinacoid  belong  to  the 
same  individual.  If  two  such  forms  be  still  further  twinned 
with  the  twinning  plane  the  clinodome  (Oil),  then  a  complex 
form  (Figure  205  b)  may  be  obtained  from  their  interpene- 
tration.  The  repetition  of  this  process  gives  rise  to  forms 
represented  by  Figure  205  c.  It  may  be  observed  that  the 
first  type  of  twin  simulates  the  symmetry  for  the  rhombic 
system,  the  second  the  symmetry  of  the  tetragonal  system, 
and  the  last  the  symmetry  of  the  cubic  system.  Sometimes 
on  crystals  of  phillipsite  the  faces  are  so  extended  as  a  result 
of  growth  that  complex  forms  similar  to  the  rhombic  dode- 
cahedron are  obtained. 

Mallard  was  so  impressed  with  the  possibilities  of  produc- 


CRYSTAL    AGGREGATES  155 

ing  forms  of  high  symmetry  by  the  combination  of  forms 
of  lower  symmetry  as  to  suggest  that  all  substances  in  their 
ultimate  molecular  structure  were  of  the  very  lowest  degree 
of  symmetry  and  that  the  apparent  higher  symmetry  is 
always  pseudo-symmetry,  being  the  result  of  twinning. 


CHAPTER   XV 

IRREGULARITIES    OF   CRYSTAL   SURFACES 

In  the  previous  pages  crystals  have  been  illustrated  and 
described  in  their  idealized  form.  Among  crystals,  as  else- 
where, the  ideal  is  very  different  from  the  real  and  is  at  best 
only  approached  by  a  relatively  small  proportion  of  the 
crystal  individuals.  It  has  already  been  pointed  out  that 
distorted  crystals  are  exceedingly  common.  This  is  due  to 
a  variation  in  central  distance  among  the  faces  composing 
the  crystal  form.  The  crystal  drawings  contained  in  the 
previous  chapters  are  almost  entirely  of  such  a  character  as 
to  represent  only  idealized  crystals.  There  are  other  irregu- 
larities in  regard  to  crystals  which  have  not  been  acknowl- 
edged in  our  previous  descriptions. 

Curved  Faces.  Although  crystals  have  been  assumed  to 
be  always  bounded  by  plane  surfaces  there  are  nevertheless 
many  substances  whose  crystal  faces  are  somewhat  curved. 
This  has  been  frequently  observed  on  diamond  (Figure  206), 


Fig.    206  —  Curved    surfaces    ex-          Fig.  207  —  Curved  crystal  of  sid- 
hibited  by  diamond.  erite. 

siderite  (Figure  207),  dolomite  and  calcite.  The  cause  of  the 
curvature  of  crystal  faces  has  in  some  instances  been  ex- 
plained as  a  result  of  solution  or  corrosion  which  has  blunted 
the  edges  and  corners  by  dissolving  the  substances  from  the 
edges  and  corners  more  rapidly  than  from  the  central  parts 


IRREGULARITIES  OF  CRYSTAL  SURFACES  157 

of  the  faces.  This  is  the  explanation  advanced  by  Gold- 
schmidt  for  the  curvature  commonly  exhibited  on  crystals 
of  diamond.  The  same  explanation  probably  holds  for 
apatite  and  calcite.  In  other  instances  the  corners  and  edges 
of  the  crystal  are  sharper  and  more  prominent  than  in  the 
normal  crystal.  This  can  hardly  be  ascribed  to  corrosion  of 
crystals  once  normal,  since  the  general  effect  of  corrosion 
is  to  reduce  and  to  render  less  prominent  the  outstanding 
points  and  edges  of  the  crystal.  Forms  such  as  that  repre- 
sented for  siderite  are  apparently  original  forms  assumed 
by  the  substances  on  crystallizing.  The  curvature  may  be 
due  to  the  aggregation  of  sub-individuals  in  positions  of 
approximate  parallelism. 

Vicinal  Planes.  In  accordance  with  the  law  of  rationality 
of  axial  parameters,  only  such  faces  occur  on  crystals  as  give 
rise  to  indices  which  are  small  whole  numbers  or  zero.  The 
examination  of  the  inclination  of  the  faces  found  upon 
crystals  shows  that  it  is  usually  possible  to  write  the  symbols 
of  the  various  forms  without  employing  indices  larger  than 
ten.  Notwithstanding  this  generalization  one  frequently 
observes  the  presence  of  faces  occupying  a  position  almost 
parallel  to  certain  faces  of  large  development  and  simple 
index  relationships.  The  occurrence  on  quartz  of  numerous 
faces  almost  parallel  to  the  prism  of  the  first  order  (1010) 
has  been  frequently  observed.  The  following  forms  of  this 
character  have  been_ noted:  (12.0.12.1),  (13.0.13.1),  (15.0.15.1), 
(16.0.16.1),  (17.0.17.1),  (18.0.18.1)  and  (28.0.28.1).  These 
forms  are  not  more  than  four  degrees  from  parallelism  with 
the  faces  of  the  prism  of  the  first  order.  Faces  of  this  char- 
acter are  also  well  represented  on  many  crystals  of  fluorite 
which  appear  at  first  sight  to  be  completely  enclosed  by  faces 
of  the  cube.  On  closer  examination  one  frequently  observes 
in  addition  to  the  cube  faces  very  flat  tetrahexahedra  having 
the  symbol  approximately  represented  by  (32.1.0).  In  some 
instances  a  large  number  of  these  peculiar  planes  may  be 
found  following  one  another  and  in  this  manner  rounding 
edges  and  corners  and  giving  rise  to  apparent  curvature 
of  crystal  faces.  In  contrast  to  crystal  faces  whose  indices 
are  simple  these  complex  planes  are  spoken  of  as  vicinal 
planes. 


158 


CRYSTALLOGRAPHY 


Striations.  Crystal  faces  are  frequently  striated  by  a 
series  of  parallel  lines.  A  microscopic  examination  of  such 
faces  shows  that  these  striations  are  due  to  ridges  formed  by 
the  oscillation  in  the  growth  of  the  crystal.  Cubes  of  pyrite 
are  commonly  marked  by  the  presence  of  striae  which  are 
parallel  to  the  cube  edges.  On  adjacent  faces  these  strise 
are  at  right  angles  to  one  another.  (Figure  44).  While 
certain  portions  of  the  faces  of  these  crystals  have  undoubt- 
edly the  direction  characteristic  of  cube  planes  the  major 
portion  of  the  surfaces  consists  of  a  pyritohedron.  Stria- 
tion  on  crystal  surfaces  is  exceedingly  common.  On  crystals 
of  quartz  the  prismatic  faces  are  usually  striated  at  right 
angles  to  the  vertical  axis.  In  tourmaline  many  of  the  faces 
of  the  prismatic  zone  exhibit  striations  parallel  to  the  vertical 
axis. 

Etching  Figures.     When  a  crystal  possessing  smooth  plane 
surfaces  is  subjected  to  the  influence  of  a  corrosive    reagent 
the   solution   takes    place    in    a   peculiar 
manner.      First    of    all    a    few    pits    are 
formed  which  become  larger  as  the  cor- 
rosion is  continued  until  finally  their  walls 
coming   into    contact    with    one   another 
the  pits  disappear  and    small  elevations 
are  formed.     These  pits,  which  are  called 
etching   figures,    can   be   easily   produced 
Fig.    208  —  Etching  on  the  plane  surfaces  of  the  crystals  by 
figures  on  the  basal  pina-  using    an    appropriate    corrosive    in    the 
coid  of  albite  showing  laboratory.     Figures  208  and  209  repre- 
3t  of  twinning.      ^^  the  form  and        mmetry  of  guch  pitg 

on  a  basal  pinacoid  of  albite  and  on  certain  faces  of  axinite  re- 
spectively. The  etching  figures  betray  the  physical  symmetry 
of  the  face  on  which  they  occur.  It  has  previously  been  shown 


no  no 

Fig.  209  —  Etching  figures  on  axinite  showing  the  general  absence  of  sym- 
metry. 


IRREGULARITIES  OF  CRYSTAL  SURFACES 


159 


how  the  employment  of  this  method  may  be  utilized  in  the 
determination   of   crystal   symmetry.     Figure   210  represents 


Fig.  210  —  Etching  figures  on  the  basal  pinacoid  of  spangolite  showing 
trigonal  symmetry  of  this  plane.  —  After  Penfield. 

the  corrosion  forms  on  the  basal  pinacoid  of  the  mineral 
spangolite.  Etching  figures  may  be  observed  on  the  natural 
surfaces  of  many  minerals,  notably  on  quartz  and  topaz.  It 
is  assumed  that  these  natural  etching  figures  have  been  formed 
by  some  corrosive  acting  upon  plane  crystal  surfaces. 


CHAPTER   XVI 


CRYSTAL   DRAWINGS 

It  is  convenient  to  represent  crystals  by  means  of  models 
or  drawings.  For  certain  purposes  the  model  has  the  prefer- 
ence, while  for  compactness  and  for  publication  the  drawing 
is  essential. 

Several  different  types  of  drawings  are  used  by  crystallog- 
raphers.  One  of  the  commonest  indicates  the  appearance 
of  the  crystal  when  it  is  viewed  from  a  long  distance  and  in 
the  direction  of  some  prominent  series  of  zone  edges.  This 
type  of  drawing  is  known  as  a  plan  or  orthographic  projection. 
In  practice  the  zone  edge  which  represents  the  direction  of 
vision  for  plans  is  parallel  to  the  vertical  axis  c.  Such  a 
plan  is  bounded  by  a  series  of  lines  representing  the  edges 


\ 


Fig.  211  —  Clinographic  and  orthographic  drawings  of  a  crystal  of  wernerite. 

between  the  faces  composing  the  prismatic  zone  and  the 
terminal  faces.  The  edges  between  the  various  terminal 
faces  are  indicated  by  a  network  of  lines  within  the  bound- 
aries set  by  the  intersection  of  the  terminal  faces  with  those 
of  the  prismatic  zone.  The  plan  does  not  exactly  represent 
the  picture  which  could  be  obtained  by  a  camera  since  the 
eye  of  the  observer  is  represented  as  being  at  infinity  —  the 
lines  which  join  the  various  corners  shown  in  the  plan  to 
the  point  of  vision  being  parallel  to  one  another.  It  is  not 


CRYSTAL   DRAWINGS 


161 


possible  to  tell  from  an  examination  of  such  a  plan  whether 
the  terminal  faces  of  the  crystal  are  flat  or  steep.  This  is 
one  of  the  main  disadvantages  of  representing  crystals  in 
this  fashion.  On  the  other  hand  when  a  crystal  is  terminated 
by  a  large  number  of  faces,  several  of  which  lie  almost  in 
one  plane,  it  is  easier  to  distinguish  the  relationship  of  these 
faces  to  one  another  in  the  plan  than  in  any  other  type  of 
crystal  drawing.  Figure  21  la  represents  a  crystal  of  werne- 
rite  while  Figure  2ilb  is  a  plan  parallel  to  the  basal  pinacoid. 
The  relationship  of  the  various  terminal  faces  in  this  plan 
is  more  readily  understood  than  in  the  oblique  representa- 
tion of  the  same  crystal. 

Preparation  of  a  Plan  from  a  Gnomonic  Projection.  In  the 
gnomonic  projection  the  intersection  points  of  the  face  nor- 
mals appear  as  points.  Such  a  projection  indicates  to  us 
neither  the  magnitude  of  the  faces  on  the  crystal  nor  the 
method  of  intersection  of  the  faces  to  form  edges.  It  is 
always  possible,  however,  to  learn  from  such  a  projection  the 
direction  of  the  edge  between  any  pair  of  faces  upon  a  crystal. 
The  direction  of  a  line  representing  the  edge  between  a  pair 
of  faces  in  a  plan  parallel  to  the  plane  of  the  gnomonic  pro- 


Fig.  212  —  Sketch  illustrating  the  method  of  preparing  orthographic  and 
clinographic  projections  of  a  crystal  from  the  gnomonic  projection. 

jection  is  at  right  angles  to  the  line  connecting  the  projection 
points  of  the  two  faces.  It  is  therefore  easy  to  make  a  plan 
representing  the  faces  which  appear  on  a  gnomonic  projec- 
tion. This  may  be  conveniently  carried  out  on  part  of  the 
sheet  containing  the  projection  or  upon  another  sheet  of  paper 
definitely  fixed  upon  the  drawing  board  with  regard  to  the 


1 62  CRYSTALLOGRAPHY 

sheet  on  which  the  projection  appears  (Figure  212).  If  the 
crystal  represented  in  the  projection  be  large  it  may  be  con- 
venient to  measure  the  length  of  the  various  edges  and  to 
make  the  drawing  either  in  the  natural  scale  or  in  some 
multiple  of  it.  When  the  gnomonic  projection  is  prepared 
from  a  crystal  of  small  dimension  it  is  necessary  —  by 
use  of  the  telescope  of  the  goniometer  —  to  make  a  free-hand 
representation  of  the  relative  magnitude  of  the  faces  as  they 
appear  on  the  crystal.  In  this  case  the  proportionate  length 
of  the  edges  between  the  faces  contained  in  the  plan  are 
derived  from  the  free-hand  sketch.  In  the  preparation, 
therefore,  of  such  a  plan  it  is  necessary  to  have  information 
on  the  following  points: 

1.  The  direction  of  the  lines, 

2.  The  length  of  the  lines, 

3.  The  position  of  the  lines. 

The  direction  is  readily  obtained  from  a  gnomonic  projec- 
tion since  the  line  representing  the  edge  between  any  pair  of 
faces  is  at  right  angles  to  the  line  connecting  the  projection 
points...  The  length  of  the  edge  line  in  the  plan  is  depend- 
ent first  upon  its  relative  length  on  the  crystal,  and 
second,  upon  the  scale  in  which  it  is  desired  to  prepare  the 
drawing.  The  scale  may  be  the  same  as  that  shown  upon 
the  crystal  —  natural  scale  —  or  any  multiple  of  the  same. 
The  position  on  the  sheet  of  paper  for  the  first  line  is  arbi- 
trarily selected.  When  this  is  represented  in  its  proper  posi- 
tion the  lines  which  start  from  the  ends  of  this  first  line  are 
also  fixed.  From  the  ends  of  this  first  line  others  may  be 
drawn  and  similarly  the  various  lines  required  to  complete 
the  projection  may  be  drawn  from  the  ends  of  lines  already 
fixed. 

Clinographic  Drawings.  If  a  cube  be  viewed  from  a  point 
in  line  with  one  of  the  axes  of  reference  only  one  face  is  visi- 
ble and  it  appears  as  a  square  (Figure  213 a).  The  most 
information  is  obtained  from  drawings  which  represent  the 
crystal  when  viewed  from  a  point  lying  within  one  of  the 
octants  and  not  contained  in  a  plane  which  bounds  the  oc- 
tants. For  many  years  crystallographers  by  general  agree- 
ment have  been  accustomed  to  represent  crystals  by  drawings 


CRYSTAL  DRAWINGS 


163 


in  which  the  direction  of  vision  is  the  same  for  all  systems. 
This  direction  may  be  best  illustrated  by  reference  to  the 
cube.  If  instead  of  viewing  the  cube  in  the  manner  repre- 
sented in  Figure  21 3  a  we  revolve  it  about  its  vertical  axis 
through  a  certain  angle  (18°  26'),  we  obtain  a  picture  showing 
only  two  of  the  cube  faces  (Figure  213  b).  Since  it  is  evident 
that  this  is  not  an  advantageous  representation  of  the  cube 
or  any  other  crystal  it  has  been  agreed  to  elevate  the  eye 
9°  28'  so  as  to  obtain  a  picture  such  as  is  represented  in 
Figure  44.  The  picture  shows  three  of  the  cube  faces  and 
represents  the  standard  orientation  for  clinographic  drawings. 
A  picture  of  this  kind  can  be  readily  prepared  starting  from 


a  b 

Fig.  213  —  The  cube  as  seen  from  different  directions. 

the  gnomonic  projection  and  plan.  A  line  from  the  eye  to 
the  crystal  centre  would  intersect  the  plane  of  gnomonic 
projection  at  a  certain  point  W.P.  which  is  marked  by  a 
star  in  Figure  212.  A  plane  passing  through  the  crystal 
centre  and  at  right  angles  to  this  direction  of  vision  inter- 
sects the  plane  of  projection  in  the  line  L.L.  All  those  faces 
whose  projection  points  lie  in  front  of  the  line  L.L.  will  be 
directly  visible,  that  is,  on  the  side  of  the  crystal  toward 
the  observer.  According  to  Goldschmidt  L.L.  is  known  as 
the  Leit  Linie,  while  the  point  W.P.  is  the  Winkel  Punkt. 
It  is  evident  that  if  the  direction  of  vision  be  changed,  that 
is,  if  a  new  Winkel  Punkt  be  selected  the  Leit  Linie  will 
also  change  and  certain  faces  may  become  visible  in  the 
picture  while  others  disappear.  Those  faces  whose  projec- 
tion points  lie  on  the  Leit  Linie  will  be  foreshortened  in  the 
picture  to  a  single  line,  while  for  the  other  faces  the  nearer 
their  projection  points  are  to  the  line  L.L.  the  more  fore- 
shortened do  they  appear.  The  point  of  vision  is  placed 


164 


CRYSTALLOGRAPHY 


at  infinity  so  that  all  parallel  edges  are  represented  by  parallel 
lines  in  the  drawing.  Having  already  prepared  a  gnomonic 
projection  and  a  plan  (Figure  212)  as  a  preliminary  we  may 
proceed  with  the  preparation  of  the  clinographic  projection. 
In  the  making  of  such  a  drawing  we  require  to  know  the  posi- 
tion, direction  and  length  of  the  lines  involved,  which  are 
fixed  in  the  following  manner: 

1.  To  find  the  direction  in  a  clinographic  projection  of  the 

line  representing  an  edge  between  two  faces  whose 
gnomonic  projection  points  are  indicated,  draw  a 
line  through  the  projection  points  and  produce,  if 
necessary,  to  intersect  the  Leit  Linie.  The  direction 
of  the  edge  in  question  is  at  right  angles  to  the  line 
connecting  this  intersection  point  with  the  Winkel 
Punkt. 

2.  The  corners  represented  in  the  plan  are  contained  in  the 

clinographic  projection  on  lines  from  the  corners  at 
right  angles  to  the  Leit  Linie.  Through  the  point 
b  draw  a  line  normal  to  the  Leit  Linie,  and  arbitrarily 
select  b'  a  point  on  this  normal  to  represent  the  point 
b.  Through  the  point  c  draw  a  normal  to  the  Leit 
Linie  and  draw  b'e'  to  represent  the  edge  be.  Sim- 
ilarly c'd',  d'a'  and  a'b'  may  be  drawn,  thus  complet- 
ing the  outline  in  the  clinographic  projection  for  the 
basal  pinacoid.  The  other  lines  involved  in  the  pro- 
jection may  be  drawn  in  the  same  way. 

3.  The  lines  representing  the  prismatic  zone  appear  in  the 

clinographic  projection  at  right  angles  to  the  Leit 
Linie.  The  direction  and  position  therefore  of  any 
one  of  these  lines  may  be  obtained  by  drawing  a 
line  at  right  angles  from  the  point  in  the  plan  repre- 
senting it. 


CHAPTER   XVII 

ILLUSTRATION    OF   CRYSTALLO GRAPHIC 
INVESTIGATIONS 

By  permission  of  several  crystallographic  friends  the 
extracts  in  this  chapter  have  been  selected  from  a  number  of 
publications  indicating  the  problems  with  which  crystallog- 
raphy has  to  deal  and  the  most  appropriate  methods  of  solv- 
ing them.  It  may  be  admitted  that  in  previous  chapters  the 
information  regarding  the  determinations  of  axial  ratios  and 
polar  elements  is  not  sufficient  to  make  it  possible  for  the 
student  to  conduct  with  confidence  many  of  the  investigations 
connected  with  the  study  of  crystals.  The  extracts  selected 
illustrate  the  method  of  procedure.  Naturally  the  greatest 
difficulty  is  met  in  connection  with  the  monoclinic  and  tri- 
clinic  systems,  where  some,  or  all,  of  the  axes  of  reference 
are  oblique. 

CUBIC  SYSTEM 

NEW    FORMS    OF    SPERRYLITE 

BY  VICTOR  GOLDSCHMIDT  AND  WILLIAM  NICOL 

(American  Journal  of  Science,  June,  1903) 

The  sperrylite  examined  by  us  during  the  summer  of  1902 
<vas  obtained  at  the  Vermillion  Mine,  Algoma  District,  Province 
of  Ontario.  This  is  the  original  locality  from  which  was  ob- 
tained the  material  examined  by  Wells  and  Penfield  1  and  that 
described  by  Walker.2  By  " panning"  the  loose  material  at 
the  mouth  of  the  old  shaft  a  heavy  residue  was  obtained. 

1  This  Journal  (3),  xxxvii,  67  et  seq.,  1889.     Zeitschr.  f.  Kryst.,  xv,  285 
u.  290,  1889. 

2  This  Journal  (4),  i,  110  et  seq.,  1896.     Zeitschr.  f.  Kryst.,  xxv,  561, 
1895. 


166  CRYSTALLOGRAPHY 

When  this  residue  was  examined  by  a  needle  and  pocket  lens 
the  brilliant  silver-white  crystals  of  the  mineral  were  readily 
separated  from  the  magnetite,  etc.,  occurring  with  them. 
The  crystals  are  exceedingly  small,  but  on  account  of  their 
brilliant  metallic  lustre  the  measurement  of  the  minute  faces 
was  easily  possible.  The  crystals  were  measured  on  the  two- 
circle  reflecting  goniometer,  making  use  of  the  reduced  signal. 
Nine  crystals  were  measured,  which  showed  the  following 
forms : 

Crystal  System  Regular  Pentagonal  Hemihedrism 

*      *  *      *  ****  *** 

Letter:  c         ag'e         hb'SkmqBpu^x 

Symb.Gdt.:        0      +  !<> -fO +$0 +|0 -fO -10     £        $•        \        \       1        \\     \\      -%\ 
Symb.  Miller:  001     103    205    102    305    203    101     114     113    112  335   111    212    214    213 

Of  these  forms,  those  designated  with  an  asterisk  are  new  for 
sperrylite.     In  addition  to  these,  two  other  forms, 

?z  =  +  H(315);  ?D  =    +iJ(326), 
were  observed,  but  not  with  absolute  certainty. 

Charles  W.  Dickson1,  doing  post-graduate  work  at  the 
School  of  Mines,  Columbia  University,  New  York,  has  suc- 
ceeded in  separating  sperrylite  crystals  from  the  unaltered 
chalcopyrite  from  the  Victoria  Mine,  in  the  Sudbury  District. 
He  reports  by  letter,  "I  found,  on  examination  of  two  of  the 
best  crystals,  the  trapezohedrons  211  and  411,  besides  the 
originally  described  forms." 

The  form  e  occurs  on  all  the  crystals  measured,  having  only 
half  the  number  of  faces  developed,  in  accordance  with  pentag- 
onal hemihedrism.  This  form  has  been  regarded  as  positive, 
and  the  positive  or  negative  character  of  the  other  forms  has 
been  determined  in  accordance  with  this.  This  criterion 
may  be  accepted  as  decisive.  On  account  of  the  minute  size 
of  the  crystals  it  was  not  possible  to  attempt  to  decide  this 
point  by  etch-figures. 

The  nine  crystals  measured  showed  the  following  combina- 
tions: 

Combinations 

Crystal  No.  ll       c.g'e.b'..mq.pu....     Fig.  2. 

Crystal  No.  2:       c..      e p 

Crystal  No.  3:       c..     e qBp 

1  This  Journal  (4),  xv,  137,  1903. 


CRYSTALLOGRAPHIC    INVESTIGATIONS 


167 


Crystal  No.  4:       c..     e.b'd.mqBpu^x..     Fig.  3. 


Crystal  No.  5: 

c  .    . 

e  . 

Crystal  No.  6: 

c  .    . 

e  h 

Crystal  No.  7: 

c  .    . 

e  . 

Crystal  No.  8: 

c  a  . 

e  . 

Crystal  No.  9: 

c  .    . 

e  . 

6k 


p  u 


(z)  . 


Fig.  4. 
Fig.  5. 


6  k  m  q  .    p 

P 

.    .    .     q     .  p 


.(D) 


From  this  table  of  combinations  the  relative  frequency  and 
importance  of  the  individual  forms  can  be  observed. 


(B) 


Fig.  1 


Fig.  2 


Fig.  3 


The  following  arrangement  shows  the  relative  importance: 

c  always  present,  usually  predominating. 

p  almost  always  present,  usually  smaller  than  c,  sometimes 
of  equal  importance  with  c,  now  and  then  predominating. 

e  always  present,  but  less  important  by  far  than  p. 

q  5  u  m  very  small,  but  occurring  not  seldom. 

a  g*  h  b*  k  B  x  occurring  seldom  and  unimportant. 

z  D  scarce,  faint  and  uncertain. 

Up  to  the  present  time,  from  the  measurements  of  Pen- 
field  and  Walker,  the  following  forms  were  known: 

c  =  0  (001);  5  =  10  (101);  e  =  +  J  0  (102);  p  =  1  (111). 

These  are,  in  fact,  the  most  important  forms.  In  addi- 
tion Walker  gives  the  form  it  (5.2.10),  determined  from 
measurements  of  plane  angles.  This  form  was  not  found  on 
any  of  the  nine  crystals. measured.  It  lies  in  the  zone  +  J  q, 
between  e  =  J  0  and  q  =  J,  —  an  important  zone  for  this 
mineral.  It  may  perhaps  be  identified  with  the  form  \f/  =  H- 
In  the  same  zone  lies  the  uncertain  form  D  =  -J-J-  and  the 
well  defined  form  u  =  |  1.  Figures  2,  3,  4  and  5  show  crystals 
Nos.  1,  4,  5  and  6.  Figure  1  gives  an  ideal  picture  of  all 
the  forms.  Owing  to  the  difficulty  of  drawing  so  many 


168 


CRYSTALLOGRAPHY 


narrow  faces  and  reproducing  them  by  lithographic  plate, 
only  part  of  the  crystal  is  shown.  The  forms  x  and  h  are 
wanting,  as  their  existence  was  decided  with  certainty  by 
measurements  made  after  the  drawing  was  prepared.  In 
Figures  2-5  the  attempt  has  been  made  to  preserve  the 
proportions  of  the  faces  as  they  actually  occur  in  nature. 

The  measurement,  or  rather  the  discussion  of  the  results  of 
the  measurement,  offered  several  difficulties  which  can  be  re- 
ferred to  the  small  size  of  the  crystals  and  especially  of  the 
subordinate  faces.  These  faces  are^  not  only  absolutely  but 
also  relatively  small,  that  is,  when  compared  as  to  size,  with 
the  principal  faces,  c  and  p,  they  stand  far  in  the  background. 
This  may  be  readily  seen  from  an  inspection  of  the  figures. 
In  discussing  the  measurements  in  the  gnomonic  projection, 


Fig.  4 


Fig.  5 


the  following  unexpected  phenomenon  was  observed.  In 
cases  where  the  principal  faces  c  and  p  occupied  their  exact 
position,  showing  that  the  crystal  had  not  been  disturbed  in 
its  growth,  yet  the  signal  reflections  from  the  subordinate 
faces  were  displaced  in  many  ways,  and  this,  too,  when  these 
faces  were  well  defined  as  to  lustre  and  perimeter.  The 
minuteness  of  the  subordinate  faces  is  exceptional.  It  is 
interesting  to  note  what  minute  faces  can  be  measured  by 
the  excellent  instruments  of  the  present  day.  The  following 
measurements  were  obtained  under  the  microscope: 


Crystal  No.  4,  Fig.  3. 
The  large  upper  c  face, 
The  large  front  p  face, 
The  upper  e  face,  to  the  right  from  p,        " 
The  lower  ^  face,  to  the  right  from  p,         " 
The  lower  x  face,  to  the  left  from  p,  " 

The  lower  6  face,  length  =  0.3 


longest  diameter  =  0.8  nan 

=  0.6 

"  "         =  0.06 

=  0.03 
"         =  0.03 
;  width   =  0.015 


CRYSTALLOGRAPHIC  INVESTIGATIONS  169 

The  displacements  and  consequent  variations  in  the  case  of 
such  small  faces  may  have  two  causes: 

1.  Optical  displacement  and  diffusion  of  the  signal  reflec- 
tion owing  to  bending  of  the  light.     The  smaller  the  face  the 
stronger  is  this  effect  and  the  less  the  certainty  in  determining 
the  position  of  the  face. 

2.  Curvature  of  faces  near  the  outlines.     This  is  a  con- 
sideration of  importance  in  connection  with  the  origin  of  the 
crystal  and  demands  special    study.     Sperrylite    itself    may 
perhaps  furnish  suitable  material  for  this  on  account  of  its 
perfect  metallic  lustre,  which  makes  it  possible  to  secure  good 
signal  reflections  from  even  minute  faces:    on  account  of  the 
simplicity   of   the   crystal   system   and   the   certainty   of   the 
elements    and    calculated    angles:     on   account   of    the   perfect 
formation  of  the  principal  faces,  the  great  variety  of  kinds 
of    faces   and   the   large   number   of  individual   faces,    which 
are   incident   to   the   regular   system.     It   may   be   said:    the 
degree  of  perfection  of  a  face  is  a  function  of  its   absolute  and 
relative   size.      Very   large   faces   are   usually   imperfect   since 
their  parts  and  the  parts  of  the  layers  underneath  have  a 
development  history  exposed    to    different    influences.       The 
faces    of   smaller  crystals  are,  as  a  rule,  better  formed  than 
those  of  larger  crystals.     Below  a  certain  size  the  faces  again 
become    more    imperfect.     This    may    be    assigned    to    two 
causes: 

1.  The  relatively  small  faces  are  those  derived  from  higher 
complication.     They    are    weak    and    more    easily    displaced. 
They  are  mobile,  like  the  leaves  and  branches  in  the  wind, 
while  the  trunk  is  unmoved. 

2.  In  the  case  of  small  faces  the  marginal  parts  are  rela- 
tively  large.     The   marginal   parts,    however    (on   edges   and 
corners),  are  the  places  at  which  the  particles  are  bounded 
only  on  one  side  by  securely  situated  particles  which  are,  as 
well,  oriented  themselves  and  have  an  orienting  influence  on 
their  neighbors.     This  is  the  place  of  the  manifold  formation 
of  varying  resultants  by  complication.1     The  midfield  of  not 
too  great  a  face  is  therefore   usually  the  smoothest.     Con- 
vexity readily  occurs  at  the  perimeter  and  corners.     Small 

1  Compare  Zeitschr.  f.  Kryst.,  xxix,  47,  1897. 


170  CRYSTALLOGRAPHY 

faces  lying  close  together  and  very  little  inclined  towards 
one  another  frequently  shade  into  each  other  by  round- 
ing. It  may  happen  that  only  the  border  part  is  present,  no 
smooth  middle  part  has  been  formed.  If,  now,  the  border 
part  is  large  relatively,  it  furnishes  much  light  for  the  signal 
reflection.  In  this  way  the  signal  reflection  from  the  inner 
portion  of  the  face,  when  this  is  small  and  of  low  brilliancy, 
may  become  darkened  and  eclipsed  by  that  of  the  border 
parts.  If  the  faces  are  too  small  it  is  not  possible  to  shut 
off  the  border  portions  by  means  of  the  shutter  at  the  eye- 
piece of  the  telescope.  It  is  not  advisable  at  this  place  to 
go  more  deeply  into  these  interesting  points.  They  should 
form  the  ground  of  a  special  investigation. 

The  discussion  of  the  position  of  the  faces  and  the  symbols 
of  the  subordinate  ones  must  be  undertaken  with  caution  on 
the  above  mentioned  grounds.  Considerable  experience  is 
necessary  in  order  to  be  able  to  separate  here  the  certain  from 
the  uncertain.  One  must  also,  by  measuring  numerous  crys- 
tals, accustom  one's  self  to  the  peculiarities  of  the  kind 
of  crystal  at  hand.  On  the  one  hand,  there  is  the  tempta- 
tion to  symbolize  the  positions  of  the  diffused  signal  reflec- 
tions. This  may  lead  to  great  confusion.  On  the  other  hand, 
there  is  the  danger  of  demanding  too  great  precision,  and 
therefore  of  finding  here  only  the  principal  forms  c,  p,  e. 
It  was  attempted  by  careful  discussion  of  the  measurements 
and  their  picture  in  the  gnomonic  prpjection  to  obtain  as 
many  certain  and  typical  forms  as  possible.  The  forms 
here  given  as  certain  may  be  accepted.  Those  indicated  by 
(?)  viz.,  z  and  D,  possess  a  high  degree  of  probability.  Under 
these  circumstances  a  greater  variation  than  usual  must  be 
allowed  between  the  positions  of  the  faces  as  determined  by 
measurement  and  by  calculation. 


CRYSTALLOGRAPHIC  INVESTIGATIONS  171 

HEXAGONAL  SYSTEM 

APATITE   FROM   MINOT,    MAINE 

BY  JOHN  E.  WOLFF  AND  CHARLES  PALACHE 

(Proc.  Am.  Academy  of  Arts  and  Sciences,  March,  1902) 

In  the  summer  of  1901,  while  prospecting  for  tourmaline 
or  other  gem  minerals  on  the  farm  of  Mr.  P.  P.  Pulsifer  in 
Minot,  Maine,  a  pocket  was  opened  in  the  granite  containing 
the  material  here  described.  It  was  first  brought  to  our 
notice  by  Mr.  C.  L.  Whittle,  formerly  of  this  Department, 
and  the  whole  was  subsequently  acquired  by  the  Harvard 
Mineralogical  Museum. 

This  find  is  noteworthy  for  the  unusually  rich  purple  color 
of  the  crystals,  and  the  purity,  crystalline  perfection,  and 
abundance  of  the  material,  which  comprises  about  two 
thousand  loose  crystals  or  fragments  of  crystals  with  a  total 
weight  of  over  a  kilogramme,  and  about  a  dozen  large  groups 
of  crystals  on  the  matrix.  Of  the  loose  crystals  about  three 
hundred  show  at  least  one  perfect  termination,  five  hundred 
are  slightly  less  perfect,  and  the  rest  imperfect  or  fragmentary. 

CRYSTALLOGRAPHY  * 

The  apatite  crystals  are  in  general  of  pronounced  prismatic 
habit,  the  average  size  being  about  1  cm.  in  height  and  0.5 
cm.  in  diameter.  Crystals  larger  than  this  are,  however, 
common,  the  largest  measuring  nearly  3  cm.  in  height  and 
diameter.  Crystals  smaller  than  the  average,  which  are  also 
numerous,  tend  to  assume  a  more  or  less  rounded  habit  by 
nearly  equal  development  of  prismatic  and  terminal  planes. 

The  crystals  are  generally  so  implanted  upon  a  terminal 
face  that  one  end  has  developed  freely,  and  the  fact  that  over 
three  hundred  loose  crystals  with  complete  single  termina- 
tion and  prism  zone  were  obtained  from  the  collection  shows 
how  prevailing  is  this  habit  of  growth.  Occasionally  the 
attachment  to  the  matrix  is  by  a  prism  plane,  and  then  both 
terminations  are  developed. 

1  By  C.  Palache. 


172  CRYSTALLOGRAPHY 

The  forms  observed  were  as  follows,  the  letters  used  being 
those  of  Dana: 
c  (0001),  m  (lOlO),  a  (1120),  h  (2130),   z   (3031),  y_(2021), 

x  (1011),  r  (1012),  w  (7073),  s  (1121),  M  (2131),  Mi  (3121). 

Four  crystals  were  carefully  measured  on  the  two-circle 
goniometer  and  the  same  forms  found  on  all.  The  results  of 
measurement  of  the  better  developed  forms  agreed  so  well 
among  themselves  that  it  seemed  worth  while  to  calculate  the 
axial  ratio  from  the  better  readings,  and  this  was  done,  using 
the  forms  y,  x,  r  and  s.  The  following  table  shows  the  average 
angle  to  the  base  from  each  of  these,  the  ratio  calculated  for 
each  crystal  and  the  average  ratio  obtained: 

Angle  from      No.  of    ,  l  Angle  from_   No.  of   ,  l 

0001  to  2021.    Faces    (  0001  to  1021.    Faces 

Cryst.  1  .  .  59°29'  5        3'  Cryst.  1  .  .  23°00'  6  1' 

"      2  .  .  59°29f  52'  "      2  .  .  23°00'  5  5' 

"      3  .  .  59°30£'  63'  "      3  .  .  23°00'  3  0' 

"      4  .  .  59°28&'  53'  "      4  .  .  22°59f  4  3' 

Angle  from  Angle  from 

0001  to  1011.  0001  to  1121. 

Cryst.  1  .  .  40°18'  6        3'  Cryst.  1  .  .  55°45'  6  1' 

"      2  .  .  40°19'  55'  "       2  .  .  55°46'  6  3' 

"      3  .  .  40°19'  56'  "      3   .  .  55°46'  4  3' 

"      4  .  .  40°18'  52'  "      4  .  .  55°45'  6  4' 

Crystal  1,  from  23  measurements,  p0  =  0.848307 

Crystal  2,  from  21  measurements,  p0  =  0.848739 

Crystal  3,  from  18  measurements,  p0  =  0.848753 

Crystal  4,  from  20  measurements,  p0  =  0.848148 

Average  from  82  measurements,  p0  =  0.848476  or  a:  c  =  1  :  0.734800 

Angle  calculated  from  p0  =  0.848476,  0001  to  2021  59°  29'  22" 

0001  to  1011  40°  18'  50" 
0001  to  1012  22°  59'  19" 
0001  to  1121  55°  45'  59" 

Two  types  of  combinations  may  be  distinguished  among 
these  crystals.  One  of  these  is  represented  in  Figure  1,  and 
consists  essentially  of  the  prism  of  the  first  order  and  the  base, 
the  edges  modified  by  narrow  planes  of  the  forms  a,  s,  r,  x 
and  y.  Crystals  of  this  type  are  not  uncommon  and  often 

1  d  is  the  difference  in  minutes  between  largest  and  smallest  readings 
for  faces  of  any  form. 


CRYSTALLOGRAPHIC    INVESTIGATIONS 


173 


show  double  terminations.  They  merge,  however,  by  slight 
gradations  into  the  second  type,  more  characteristic  for  the 
locality,  shown  in  Figures  2  and  3.  Here  the  pyramidal 
planes  become  more  prominent  and  the  most  notable  feature 
is  the  simultaneous  occurrence  of  the  right  and  left  third 
order  pyramids,  giving  the  appearance  of  the  normal  dihexag- 
onal  pyramid. 


Figs.  1  to  4 

The  different  forms  may  be  characterized  as  follows: 
c  (0001)  always  present,  generally  large,  brilliant,  and  plane 
giving  perfect  reflections. 


174  CRYSTALLOGRAPHY 

m  (lOlO)  always  present,  generally  dominant,  brilliant, 
and  generally  plane  but  sometimes  faintly  striated  vertically. 

a  (1120)  generally  present  but  narrow  and  commonly  dull 
from  deep  striation,  the  striae  vertical  and  bounded  by  faces 
of  adjoining  planes  of  m.  Occasionally  the  striations  stop 
abruptly  in  the  centre  or  near  the  boundaries  of  a  face  as 
shown  in  Figure  3,  or  they  may  be  wholly  lacking,  in  which 
case  the  face  is  brilliant  and  gives  good  reflections. 

h  (2130)  rarely  developed  and  then  narrow  as  shown  in 
Figure  4.  Surface  plane,  not  involved  in  striations  on  a. 

r  (10l2),  x  (1011)  and  y  (2021)  all  nearly  always  present 
with  all  their  faces,  in  varying  proportions  and  often  large, 
faces  always  brilliant  and  free  from  striations,  giving  perfect 
reflections. 

w  (7073)  observed  but  once  as  a  line  face  in  the  zone 
between  y  and  m. 

z  (3031)  generally  present  only  as  a  deeply  striated  face, 
sometimes  very  large  as  in  Figure  4,  giving  no  reflection  but 
determined  by  its  zonal  relation  to  /z  and  /*i.  The  striae 
bounded  by  faces  parallel  to  adjoining  planes  of  m  and  y. 
Narrow  faces  of  z  giving  faint  reflections  sometimes  present 
on  the  edges  of  the  striae  nearest  to  m. 

s  (1121)  always  present  with  brilliant  faces,  often  large. 

IJL  (2131)  and  ^i  (31  21)  are  both  present  on  many  crystals, 
but  vary  widely  in  size,  quality  and  regularity  of  develop- 
ment. Generally  the  faces  of  both  are  dull  and  the  forms 
are  then  indistinguishable.  On  some  crystals  their  faces  are 
brilliant  and  reflecting  but  grooved  or  pitted,  and  a  constant 
difference  in  the  character  of  these  markings  was  found  by 
which,  whea  they  were  not  too  far  developed,  the  two  forms 
could  be  distinguished.  On  /z  the  markings  ordinarily  take 
the  form  of  sharp  grooves  parallel  to  the  intersection  of  m 
and  M  as  shown  in  Figures  2  and  3.  The  grooves  seem  to  be 
in  a  way  continuations  of  the  striae  on  the  faces  of  z,  for  they 
never  extend  beyond  the  intersection  of  IJL  with  that  face, 
and  are  absent  if  z  is  not  developed.  The  grooves  are 
bounded  by  faces  parallel  to  adjacent  planes  of  s  and  of  m. 
Very  often  they  stop  short  in  the  middle  of  the  face  as  shown 
in  Figure  3. 

On  /ii  the  markings  are  in  the  form  of  irregular  pits  or  curv- 


CRYSTALLOGRAPHIC  INVESTIGATIONS  175 

ing  grooves,  sometimes  showing  approximate  parallelism  to 
the  intersection  of  m  and  /^i  but  with  an  irregularity  giving 
them  a  character  wholly  different  from  the  lines  on  /z.  No 
constant  difference  could  be  observed  in  the  brilliancy  of  the 
reflecting  portions  of  faces  of  the  two  forms,  nor  in  their 
relative  size.  Both  are  irregular  in  their  occurrence  on  indi- 
vidual crystals,  lacking  nearly  always  some  of  their  faces. 
As  shown  in  the  figures,  both  may  present  on  the  same  crys- 
tal faces  of  very  unequal  size  which  in  some  cases  are  so 
large  as  to  dominate  the  termination  of  the  crystal. 

The  occurrence  of  third  order  pyramids  in  apparently 
holohedral  combination  has  been  observed  on  apatite  from 
various  localities,  notably  Knappenwand,  Tyrol,1  Ala,  Pied- 
mont 2  and  Elba.3  But  in  none  of  the  crystals  described  does 
there  appear  to  have  been  any  observable  difference  between 
the  faces  of  the  right  and  left  forms  by  which  they  could  be 
distinguished. 

Reference  has  been  made  in  the  preceding  pages  to  stria- 
tions  which  appear  quite  constantly  on  certain  faces  of  the 
apatite.  They  are  a  striking  feature  of  the  crystals  and  the 
attempt  has  been  made  to  reproduce  them  in  the  drawings. 
Their  most  pronounced  development  was  on  the  largest  crystal 
of  the  collection,  which  is  reproduced  in  Figure  4;  the  stria- 
tions  on  the  faces  of  z  and  of  a  were  almost  equally  strong 
and  gave  the  crystal  a  curiously  tetragonal  aspect  when 
inspected  casually.  On  both  of  these  forms  the  striations  are 
doubtless  growth  forms,  the  result  of  oscillatory  combination, 
on  a  of  adjacent  faces  of  m,  and  on  z  of  planes  of  m  and  y. 
The  markings  on  the  faces  of  /x  and  MI  seem  to  have  a  different 
character,  however.  The  irregularity  of  their  development, 
appearing  on  some  faces  as  mere  grooves  or  pits,  on  others 
invading  the  whole  face  and  reducing  it  to  a  dull  surface, 
indicates  that  they  are  rather  the  result  of  etching  by  some 
agent  which  has  attacked  the  crystals  after  they  were  formed. 

1  C.  Klein,  Neues  Jahrb.  Miner.,  1871,  485;    1872,  121. 
2G.  Struever,  Att.  Ace.  Torino,  3,  125,  1867;   6,  363,  1871;   Rendic.  R. 
Ace.  Lincei,  Roma,  1899,  8  (1),  427-434. 

3  E.  Artini,  Rendic.  R.  Ace.  Lincei,  Roma,  1895,  4  (2),  259. 


176 


CRYSTALLOGRAPHY 


RHOMBIC  SYSTEM 

NOTES    ON    GOETHITE1    (Abstract) 

BY  V.  GOLDSCHMIDT  (Heidelberg)  AND  A.  L.  PARSONS  (Toronto) 
(American  Journal  of  Science,  March,  1910) 

In  the  summer  of  1908  Mr.  Parsons  collected  some  speci- 
mens of  goethite  which  occurs  in  veins  in  carboniferous  shale 
at  Walton,  N.S.  The  veins  are  brecciated  and  the  centre 
filled  with  calcite.  When  the  calcite  was  dissolved  in  hydro- 
chloric acid  the  goethite  was  left  as  a  druse  of  bright  crys- 
tals. Three  crystals  were  measured.  The  observed  forms  are  : 

b  =  Ooo   (010); 

M  =  2  oo   (210); 
a  =   °oO  (100); 
y  =  «    (HO); 
XX  =   co  4  (140); 


e  =  01 
u  =  10 

P  =  1 

p  =  31 


(Oil); 
(101); 

(in); 

(311). 


The  form  designated  as 
XX  appears  in  all  the 
crystals  as  the  face  with 
the  greatest  development. 
This  face  is  striated  length- 
wise and  is  not  a  single 
face  but  a  transition  face 
or  "Scheinflaeche,"  and  it 
gives  a  bright  band  of  light 
in  the  prism  zone  with  the 

angle  </>  varying  from  3°  to  20°.  In  this  band  are  bright  points 
which  imperfectly  indicate  the  positions  of  the  faces  °°3 
(130),  co4  (140),  005  (150),  006  (160),  co8  (180),  »12  (1.12.0). 
The  harmonic  discussion  of  the  characteristic  indices  in  this 
series  gives  003,  oo4,  006,  and  008  as  the  points  of  best 
position,  but  no  characteristic  letter  is  assigned  to  them  until 

1  Abstract  of  a  paper  in  Groth's  Zeitschrift  fiir  Krystallographie. 


Figs.  1  and  2 


CRYSTALLOGRAPHIC  INVESTIGATIONS  177 

they  have  been  determined  by  single  distinct  reflecting  faces. 
The  symbol  XX  is  used  for  the  series  and  in  Figure  2,  o>4 
represents  the  series. 

For  comparative  purposes  three  crystals  from  Lostwithiel, 
Cornwall,  were  measured  and  the  habit  and  faces  are  shown 
in  Figure  1.  The  observed  forms  are  b  =  0  °°  (010);  M  = 
2oo  (210);  x  =  ^oo  (430);  y  =  oo  (HO);  1=  co2  (120); 
u  =  10  (101);  e  =  01  (Oil);  p  =  1  (111);  w  =  li  (413). 
All  the  forms  except  x  were  present  on  every  crystal.  The 
new  form  x  =  |  <»  (430)  was  present  on  two  crystals  and  for  the 
best  reflections  gave  angle  0  55°  6'  and  55°  33'.  A  poor  reflec- 
tion gave  one  face  as  57°  27'.  Angle  p  was  in  every  case  90°. 
The  calculated  angles  are  <£  55°  26',  p  =  90°.  The  face  is 
small  but  well  defined,  and  may  be  regarded  as  well  estab- 
lished. The  form  w  =  fi  (413)  is  also  new  and  is  present 
on  all  three  crystals  with  four  faces  on  each.  The  faces 
reflect  well  and  give  a  distinct  signal  cross,  but  it  is  worthy 
of  remark  that  the  reflection  is  slightly  yellow  while  the  others 
are  white.  The  angles  agree  well  among  themselves,  but  in 
every  case  the  angle  p  is  20'  to  69'  less  than  the  calculated 
angle.  The  calculated  angles  are  <£  =  77°  4',  p  =  42°  6'.  Con- 
sidering the  good  character  of  the  reflection  and  the  sharp- 
ness of  the  faces,  this  difference  is  not  easily  understood,  but 
the  form  may  be  considered  as  well  established. 

Heidelberg,  Aug.  14,  1909. 

MONOCLINIC  SYSTEM 
COLEMANITE  FROM  SOUTHERN  CALIFORNIA 

A  DESCRIPTION   OF  THE   CRYSTALS  AND  OF  THE  METHOD   OF   MEAS- 
UREMENT WITH  THE  TWO-CIRCLE  GONIOMETER 

BY  ARTHUR  S.  EAKLE 

(Bulletin  of  the  Department  of  Geology,  University  of  California,  1902) 
DESCRIPTION  OF  CRYSTALS 

Introduction.  —  The  mineralogical  collection  of  the  Univer- 
sity of  California  includes  several  excellent  specimens  of 
colemanite,  showing  beautiful  crystals  in  the  druses  and 
geodes  of  the  massive  material,  and  the  abundance  of  good 


178  CRYSTALLOGRAPHY 

measurable  crystals  afforded  an  opportunity  to  make  a  very 
complete  crystallographic  study  of  this  borate.  The  suite  of 
crystals  selected  for  measurement  included  probably  all  of 
the  habits,  and  possibly  all  of  the  forms,  possessed  by  the 
mineral. 

The  measurements  were  made  with  the  two-circle  (Zwei- 
kreisige)  goniometer,  designed  by  Goldschmidt.  Since  the 
method  of  measurement  with  this  instrument  and  the  gno- 
monic  projection  are  as  yet  not  generally  understood  by 
American  readers,  a  detailed  statement  of  the  work  follows 
the  description  of  the  crystals,  which  will  make  clear  the 
steps  used  in  the  calculation  and  projection  of  the  forms. 

Colemanite  was  first  discovered  in  Death  Valley,  Inyo 
County,  California,  in  1882  .and  in  the  following  year  the 
more  extensive  deposits  were  found  in  the  Calico  District, 
about  five  miles  from  Daggett,  San  Bernardino  County. 
Most  of  the  fine  geodal  specimens  are  from  the  Calico  Dis- 
trict and  it  is  presumed  that  the  crystals  described  here  are 
from  this  locality,  as  the  labels  indicate. 

The  crystal  forms  of  colemanite  were  first  described  by 
Jackson.1  His  description  of  the  crystals  was  quite  complete 
and  forms  the  basis  of  what  is  at  present  known  regarding 
their  forms.  Some  of  his  crystals  were  from  the  Death 
Valley  deposits,  but  the  majority  came  from  the  Calico 
District. 

Others  who  have  measured  the  crystals  are:  Hiortdahl,2 
who  published  a  short  description  of  the  forms  and  optical 
properties,  with  a  chemical  analysis  of  a  few  crystals  from 
Death  Valley;  Bodewig  and  vom  Rath,3  who  likewise  de- 
scribed some  of  the  Death  Valley  crystals,  and  also  gave  an 
account  of  the  origin  of  the  deposits;  and  Arzruni,4  who 
described  one  crystal. 

The  crystals  commonly  line  geodal-shaped  cavities  and 
some  are  quite  large,  one  in  the  collection  having  a  width  of 
ten  centimetres.  They  are  colorless  transparent,  to  white, 

1  A.  W.  Jackson.     Bull,  of  Cal.  Acad.  Sciences,  1885,  No.  2,  2-36,  and 
1886,  No.  4,  358-365. 

2  Th.  Hiortdahl.     Zeitschrift  fur  Krystallographie,  1885.     10,  25-31. 

3  C.  Bodewig  und  G.  vom  Rath.     Idem,  179-186. 

4  A.  Arzruni.     Idem,  272-276. 


CRYSTALLOGRAPHIC  INVESTIGATIONS  179 

although  a  few  are  stained  yellowish  by  iron.  Most  of  them 
are  attached  to  the  matrix  by  one  end  of  the  vertical  axis, 
leaving  one  end  well  terminated  by  front  and  rear  forms. 
This  circumstance  rendered  but  one  mounting  on  the  gonio- 
meter usually  necessary  in  order  to  measure  all  of  the  forms 
on  a  crystal,  and  measurements  consequently  could  be  rapidly 
made.  Complete  measurements  were  made  of  thirty  crystals, 
and  many  more  were  examined  for  additional  forms. 

Elements.  —  The  axial  ratio  and  angle  0  generally  accepted 
for  colemanite  are  those  determined  by  Jackson.  He  derived 
his  elements  from  the  measurements  of  one  crystal,  but  sub- 
sequently verified  his  calculations  by  the  examination  of  more 
crystals.  By  the  two-circle  method  of  measurement  all  of 
the  readings  can  enter  into  the  computation  of  the  axial 
lengths;  consequently  a  ratio  obtained  by  using  a  large 
number  of  readings  from  the  best  faces  is  presumably  more 
exact  than  when  calculated  from  a  few  interfacial  angles. 
As  shown  later  in  the  detailed  description  of  the  work,  the 
axial  ratio  and  angle  0  obtained  for  colemanite  by  the  writer 
were  as  follows: 

a:  b:  c  =  0.7768:  1:  0.5430;  0  =  110°  1'. 

For  comparison  the  elements  calculated  by  others  are  as 
follows : 

Jackson,  a:  b:  c  =  0.7755:  1:  0.5415;  ft  =  110°13'. 

Hiortdahl,  a:  b:  c  =  0.7747:  1:  0.5418;  ft  =  110°13'. 

Bodewig  and  vom  Rath,   a:b:c  =  0.7759:  1:  0.5416;  ft  =  110°16f. 

Forms. —  The  number  of  forms  observed  was  forty-seven, 
of  which  thirteen  were  new.  The  forms  are  arranged  in 
three  columns  below,  those  in  the  last  column  being  new. 
The  lettering  is  the  same  as  that  given  by  Dana  in  his  "Sys- 
tem of  Mineralogy."  Goldschmidt's  and  Miller's  symbols 
are  given  for  each  form. 

The  commonest  forms  are  c,  b,  a,  m,  t,  a,  K,  h,  0,  y,  v,  d, 
and  o.  Less  common,  yet  of  quite  frequent  occurrence,  are  k, 
X,  i,  a,  co,  e,  i/',  x,  U.  The  remainder  of  the  known  forms  are  rare: 
Q  occurred  four  times,  7  three  times,  r,  h,  V,  twice,  and 
e,  q,  B,  ©,  each  once.  W  is  common  on  one  type  of  the 
crystals,  but  occurred  only  twice  in  the  others. 


180 


CRYSTALLOGRAPHY 


Symbol. 

Symbol. 

Letter 

Gdt. 

Miller. 

Letter 

Gdt. 

Miller. 

c 

0 

001 

k 

+  31 

311 

b 

Ooo 

010 

C 

+  10.1 

10.1.1 

a 

ooQ 

100 

e 

+  12 

121 

t 

2oo 

210 

a 

+    3 

331 

m 

CO 

110 

CO 

+  13 

131 

z 

oo2 

120 

y 

-  1 

111 

H 

co3 

130 

V 

-  2 

221 

K 

01 

Oil 

Q 

-  3 

331 

a 

02 

021 

B 

-41 

411 

X 

+  20 

201 

® 

-31 

311 

V 

+  10 

101 

o 

-21 

211 

i 

-  10 

101 

y 

-  32 

321 

h 

-20 

201 

€ 

-23 

231 

W 

-  30 

301 

d 

-  12 

121 

t 

-40 

401 

Q 

-  24 

241 

U 

-60 

601 

X 

-  13 

131 

ft 

+  1 

111 

r 

-1? 

232 

Letter 

Symbol. 

Gdt. 

Miller. 

/ 

3  oo 

310 

p 

+  30 

301 

9 

-io 

502 

f 

-  80 

801 

<t> 

+  11 

522 

P 

+  12 

142 

n 

+  14 

141 

u 

+  H 

164 

M 

+  H 

165 

V 

+  11 

232 

P 

-H 

123 

w 

-14 

182 

s 

-34 

341 

The  unit  prism  m  is  the  predominating  form,  and  the  other 
prismatic  forms  are  very  narrow.  Usually  the  rear  face  of 
a  is  much  broader  than  the  front  one;  b  is  usually  present  as 
cleavage  faces,  and  when  the  natural  face  does  occur  it  is 
small.  When  the  base  c  and  orthodome  h  are  broad,  the 
remainder  of  the  terminal  forms  are  small  and  usually  few 
in  number;  on  the  other  hand,  when  these  two  forms  are 
absent,  or  very  narrow,  the  other  terminal  forms  are  well- 
developed.  As  a  rule  the  negative  forms,  that  is,  the  upper 
rear  forms,1  are  larger  and  better  developed  than  the  positive 
ones.  A  steep  pyramid  occurs  on  a  few  of  the  crystals,  but 
the  faces  gave  very  poor  reflections.  The  best  readings  indi- 
cate the  symbols  {lO.l.l}  for  the  form,  which  are  the  same 
as  determined  by  Jackson  and  cited  as  doubtful. 

Eight  forms  given  by  Jackson  were  not  observed  by  the 
writer.  These  are  {370},  {lO.19.OJ,  {l9.19.6J,  {77l},  {412}, 
{731},  {721}  and  {711 }.  According  to  him,  the  faces  of 

xThis  usage  of  the  terms  positive  and  negative  as  applied  to  the  mono- 
clinic  system  is  different  from  that  generally  used  in  this  book.  T.  L.  W. 


CRYSTALLOGRAPHIC   INVESTIGATIONS  181 

all  these  forms  were  extremely  narrow,  and  gave  either  broad 
bands  of  light  or  no  wedge-reflections  whatever.  In  all 
probability  {370}  and  {10.19.0}  are  {l20},and  {l9.19.6J  is 
J331J.  Many  of  his  original  crystals  were  examined  by  the 
writer  for  these  forms,  but  they  could  not  be  identified. 

New  Forms.  —  With   the   exception   of    {34l},  each  of   the 
new  forms  was  observed  but  once. 

/  =  3  oo  {310}.  This  form  was  represented  by  a  narrow 
face  between  {lOOJ  and  {210 }.  The  reflection  was 
bright  and  good. 

</>  P 

Measured  76°29'  90°0' 

Calculated  76  20  90  0 

p  =  -f  30  J30l}.  This  was  a  narrow  bright  face,  but  the 
image  was  only  fair,  and  the  p-angle  varied  somewhat 
from  the  calculated  one. 

<A  P 

Measured  90°2'  69°33' 

Calculated  90  0  68  57 

g  =  —  -|0  { 502 } .  This  form  occurred  as  a  small  triangular 
face,  giving  a  good  reflection. 

4>  P 

Measured  90°7'  56°20' 

Calculated  90  0  56  12 

/  =  —  80  {801 }.  This  was  a  very  narrow  dome  face,  and 
the  reflection  was  poor.  The  face  occurred  between 
{TOO}  and  {201 }  in  the  same  zone. 

</>  P 

Measured  89°23'  80°13' 

Calculated  90     0  79  51 

<}>  =  -f-fl{522}.  This  form  occurred  as  a  narrow  face 
between  J31l}  and  {ill}.  It  gave  a  fair  reflection, 
but  the  angles  vary  somewhat  from  those  calculated 
for  the  symbols.  The  symbols  are  the  simplest  and 
probably  the  most  correct  for  the  form. 

<t>  P 

Measured  75°49'  65°46' 

Calculated  76  18  66  26 


182  CRYSTALLOGRAPHY 

p  =  +J2  {142}.  This  was  a  very  small  triangular  face 
with  a  bright  image. 

<t>  P 

Measured  34°33'  52°47' 

Calculated  34  12  52  43 

u=  +-^-|{l64}.  This  occurred  on  the  same  crystal  as 
the  preceding,  and  was  also  small  and  bright. 

</>  P 

Measured  34°23'  44°28' 

Calculated  34     8  44  32 

The  next  five  of  the  following  forms  occurred  on  the  same 

crystal. 
n  =  +  14J141J.     Two  faces  of  this  form  occurred.     They 

were  both  well  developed  and  gave  good  reflections. 

</>       \  P 

27°11'  67°53' 

Measured  2?     Q  ^  ^ 

Calculated  27     5  67  42 

/z  =  +  -g-f  J165J.  This  was  a  very  bright  small  face  which 
gave  a  good  reflection. 

0  P 

Measured  38°11'  39°40' 

Calculated  38  19  39  43 

t]  =  +  if  {232}.  This  was  a  narrow  face  with  a  bright 
reflection. 

0  P 

Measured  53°41'  54°10' 

Calculated  53  44  54     1 

P  =:  —  ^|{l23J.  This  form  was  represented  by  a  small, 
well-developed  face  which  gave  a  bright  image. 

0  P 

Measured  18°30'  21°10' 

Calculated  18     5  20  51 

s  =  —  34  {341 } .  Two  large  well-developed  faces  of  this  form 
occurred  on  this  crystal,  and  it  was  afterwards  also 
observed  on  one  other.  The  reflections  were  good. 


CRYSTALLOGRAPHIC  INVESTIGATIONS  183 

P 

70°50' 

Measured        *  ( 40  48  70  46 

70  52 
Calculated  40  40  70  45 

W=-J4{T82}.  The  edge  between  {131}  and  {llOJ 
was  replaced  by  a  narrow  face  which  was  somewhat 
rounded,  and  did  not  lie  exactly  in  the  same  zone 
with  these  two  forms.  The  symbols  are  the  nearest 
simple  ones  to  agree  with  the  readings,  although  the 
measured  and  calculated  angles  do  not  closely  agree. 

<t>  P 

Measured  0°16'  63°41' 

Calculated  0  09  64  16 

Crystal  Habit.  —  Four  quite  distinct  habits  are  noticeable, 
and  these  will  be  designated  as  Habits  1,  2,  3  and  4. 

Habit  1.  —  Crystals  of  this  habit  are  characterized  by  a 
broad  base  (001)  and  rear  orthodome  (201),  the  two  faces 
meeting  in  a  long  edge,  almost  the  width  of  the  crystals. 
The  .prism  (110)  is  long  and  almost  meets  the  base,  the  unit 
pyramid  (111)  being  merely  a  narrow  truncating  form.  The 
pyramidal  forms  are  small  and  grouped  at  the  extreme  right 
and  left  corners  of  the  crystal.  The  clinodomes  (Oil)  and 
(021)  are  often  absent.  This  habit  exhibits  the  monoclinic 
symmetry  of  the  crystals  in  a  more  pronounced  way  than  the 
other  habits  do.  This  type  of  crystal  is  seen  in  Figure  1, 
Plate  2.  The  rear  orthopinacoidal  face  (100)  is  generally 
broad  like  the  dome  (201).  The  clinopinacoid  occurs  only 
as  broad  cleavage  faces,  and  many  of  the  crystals  are  cleaved 
into  tabular  plates  parallel  to  this  form. 

Habit  2.  —  In  this  habit  the  base  and  orthodomes  are 
either  very  narrow  or  completely  wanting.  The  clinodomes 
and  rear  pyramids  are  large  and  about  equal  in  size.  The 
crystals  have  a  characteristic  pointed  appearance  at  the  ends 
of  the  orthodiagonals,  the  points  being  truncated  by  small 
natural  faces  of  the  clinopinacoid.  Most  of  the  crystals 
seem  to  be  of  this  habit.  Figure  2,  Plate  2,  shows  this  habit. 

Habit  3.  —  In  this  habit  the  front  upper  terminal  faces  are 


184  CRYSTALLOGRAPHY 

quite  small,  while  the  unit  prism  faces  and  the  orthodome 
(201)  are  large.  This  gives  a  somewhat  flattened  appearance 
to  the  crystals,  and  at  the  same  time  causes  them  to  be  pointed 
at  the  ends  of  the  vertical  axis.  This  habit  is  quite  striking, 
and  the  specimens  are  quite  suggestive  of  dog-tooth  spar. 
This  type  is  seen  in  Figure  3,  Plate  2. 

Habit  4.  —  The  crystals  of  this  habit  consist  essentially 
of  the  prism  (110)  and  a  broad  steep  orthodome  (301).  The 
combination  of  these  two  forms  makes  a  very  thin  wedge- 
shaped  crystal  with  sharp  edges.  Some  of  the  other  forms 
also  occur,  but  they  are  quite  subordinate  to  these,  and  do 
not  cause  a  variation  in  the  habit.  The  dome  (301)  has  a 
rounded  or  wavy  surface  and  seems  to  grade  into  a  still 
steeper  dome,  probably  (401)  or  (601).  The  form  (301)  is 
characteristic  of  this  habit,  and  was  only  observed  twice  on 
crystals  of  the  other  habits.  Figure  4,  Plate  2,  shows  this 
habit  with  curved  edges  resembling  spear  heads.  This  habit 
was  first  described  by  H.  S.  Washington.1 

Combinations.  — The  crystals  are  so  rich  in  forms  that  many 
of  them  have  thirty  or  more  faces.  The  various  combina- 
tions observed  were  as  follows: 

1.  c,  a,  t,  m,  AC,  a,  h,  W,  ^,  a,  /3,  y,  v,  d,  C,  o,  f. 

2.  b,  a,  t,  m,  z,  K,  a,  o-,  |9,  v,  d,  p,  u. 

3.  c,  b,  a,  t,  m,  K,  a,  h,  v,  /3,  y,  v,  d,  x,  o,  e,  w,  Q. 

4.  a,  t,  m,  h,  $,  v,  o. 

5.  6,  t,  m,  z,  y,  e,  7,  n,  s,  p,  /x,  77. 

6.  c,  6,  a,  t,  m,  K,  a,  h,  j8,  y,  v,  d,  k,  o. 

7.  c,  6,  a,  t,  m,  z,  K,  a,  h,  0,  'y,  v,  d,  x,  o,  Z. 

8.  6,  a,  t,  m,  z,  K,  a,  X,  i,  0,  y,  v,  co,  d,  o,  g,  Q. 

9.  c,  6,  t,  m,  K,  a,  <j,  0,  y,  v,  co,  d,  e. 

10.  c,  6,  a,  t,  m,  /c,  a,  X,  i,  h,  a,  0,  y,  v,  «,  d,  C. 

11.  c,  b,  a,  t,  m,  z,  K,  a,  X,  i,  h,  a,  j8,  y,  v,  d,  C,  o,  Q. 

12.  c,  6,  a,  m,  z,  K,  a,  i,  U,  /?,  y,  d,  x,  e. 

13.  c,  6,  a,  t,  m,  K,  a,  i,  h,  \f/,  U,  0,  o-,  y,  v,  e,  d. 

14.  6,  a,  t,  m,  X,  i,  TF,  y,  q,  d,  7,  s,  p. 

15.  6,  a,  £,  m,  7,  A,  ^,  t/,  0,  y,  v,  d,  o. 
±G.  c,  5,  a,  f,  m,  K,  a,  i,  /i,  0,  j/,  v,  o,  d. 
17.  6,  m,  2,  H,  K,  a,  )8,  d,  z. 

1  H.  S.  Washington.     Amer.  Journ.  of  Science  (3),  1887,  34,  281-287. 


CRYSTALLOGRAPHIC   INVESTIGATIONS  185 

18.  c,  b,  a,  t,  m,  K,  a,  ft,  ft  y,  v,  co,  d,  k,  o. 

19.  b,  t,  m,  z,  K,  U,  ft  y,  v,  co,  r,  d,  e. 

20.  6,  «,  m,  A,  If,  y,  v,  d. 

21.  c,  6,  a,  £,  m,  2,  K,  a,  o-,  ft  $/,  0,  co,  d,  e. 

22.  c,  6,  a,  «,  m,  2,  K,  a,  A,  [7,  ft  y,  v,  co,  d,  a;,  o,  e,  5,  e. 

23.  c,  6,  a,  £,  m,  2,  #,  K,  a,  X,  i,  h,  a,  ft  y,  v,  co,  d,  &,  o,  0. 

24.  6,  a,  J,  m,  a,  /i,  ^,  ?/,  v,  d. 

25.  c,  6,  a,  £,  m,  K,  a,  h,  t,  U,  ft  y,  v,  d,  o. 

26.  c,  6,  t,  ra,  K,  a,  i,  ft,  ^,  [7,  ft  t/,  v,  co,  r,  d,  e. 

27.  c,  a,  «,  m,  K,  a,  X,  i,  h,  ft  y,  v,  d,  a?,  o,  e,  7. 

28.  c,  b,  a,  t,  K,  a,  h,  t,  <7,  ft  y,  v,  q,  d,  o. 

29.  c,  6,  £,  m,  2,  AC,  a,  ft  ?/,  v,  g,  d,  e,  Q. 

30.  c,  6,  a,  m,  K,  a,  i,  ft,  ^,  y,  v,  q,  d,  o. 

Projections.  —  Figures  1-10,  Plates  2  and  3,  are  clino- 
graphic  projections  to  show  the  habits  and  some  of  the  combi- 
nations. Figure  10  is  a  drawing  of  Crystal  No.  5,  which  has 
the  five  new  forms  besides  the  two  rare  ones,  e  and  7.  The 
new  form  (341)  is  a  large  face  on  this  crystal.  About  two- 
thirds  of  the  crystal  is  broken,  otherwise  it  would  probably 
have  shown  a  rich  combination  of  forms,  as  most  of  the 
common  terminal  forms  are  missing. 

Figure  11,  Plate  3,  shows  an  orthographic  projection  on  the 
base  of  all  the  forms,  with  their  relative  predominance. 

Plate  1  exhibits  a  gnomonic  projection  of  the  forms.  The 
pole  of  the  projection,  represented  by  the  dark  circle,  is 
almost  midway  between  the  projection  of  the  base  (001)  and 
the  dome  (101),  and  since  e'  is  almost  p;0the  angles  </>  and  p 

2' 

for  the  faces  on  the  positive  side  of  the  pole  vary  only  a  few 
minutes  from  those  at  corresponding  distances  from  the  pole 
on  its  negative  side.  The  crystals  of  colemanite  therefore 
have  an  apparent  orthorhombic  symmetry,  and  the  definite 
orientation  of  some  of  the  crystals  had  to  be  determined  by 
the  direction  of  extinction  on  the  clinopinacoidal  section,  this 
direction  making  an  angle  of  about  6°  with  the  vertical  axis, 
in  the  acute  angle  ft 

The  projection  shows  the  excellent  series  of  zones  in  which 
the  forms  lie.  Most  of  the  prominent  zonal  intersections  on 
the  negative  side  of  the  pole  are  occupied  by  faces,  while 


1 86  CRYSTALLOGRAPHY 

on  the  positive  side,  several  of  these  prominent  intersections 
are  not  represented  by  forms,  as,  for  instance,  the  cross  zone 
2p0  has  only  the  one  form  (201)  in  it.  Two  of  the  negative 
pyramids,  p(123)  and  w(l82)  have  their  projections  on  the 
positive  side  of  the  pole,  and  the  latter  form  lies  almost  on 
the  first  meridian.  The  pyramidal  forms  which  are  new  are 
represented  by  two  circles,  and  the  direction  of  the  prism 
(310)  and  dome  (801)  by  double-headed  arrows.  In  this 
projection  the  prismatic  forms  necessarily  have  no  points  of 
intersection,  but  their  directions  are  shown  by  the  arrows. 

METHOD 

The  two-circle  goniometer,  with  which  the  measurements 
were  made,  has  been  fully  described  by  Goldschmidt,  together 
with  detailed  instruction  in  the  calculation  and  gnomonic 
projection  of  the  forms  based  on  measurements  with  the 
instrument,1  and  also  in  a  briefer  way  by  Palache.2 

Advantages.  --  This  goniometer  possesses  manifest  advan- 
tages over  the  ordinary  reflection-goniometer,  not  alone  in 
the  simplicity  and  greater  rapidity  with  which  forms  can  be 
calculated  from  its  measurements,  but  also  because  the  two 
angular  coordinates  definitely  locate  the  form,  and  when  these 
angles  have  been  recorded  for  every  known  form  on  crystals, 
as  Goldschmidt  has  done  in  his  "  Winkeltabellen,"  a  new  form 
can  readily  be  detected  by  comparing  its  angles  with  those 
recorded.  It  would  be  practically  impossible  to  record  all 
of  the  interfacial  angles  between  known  forms;  consequently, 
as  often  happens,  the  measured  interfacial  angle  is  not  given 
in  the  standard  works  on  mineralogy,  and  quite  frequently 
after  long  trigonometrical  calculation  the  measured  form  turns 
out  to  be  well  known.  The  instrument,  however,  can  also 
serve  for  the  measurement  of  interfacial  angles,  and,  in  the 
opinion  of  the  writer,  is  superior  for  this  kind  of  measure- 

1  V.  Goldschmidt.  —  Goniometer  mit  zwei  Kreisen.  Zeitschrift  fur 
Krystallographie,  1893.  21,  210-232.  Die  zweikreisige  Goniometer  (Modell 
1S96)  uiid  seine  Justirung.  Idem  1898,  29,  333-345. 

2  Charles  Palache.  —  On  Crystal  Measurement  by  means  of  Angular 
Coordinates  and  on  the  Use  of  the  Goniometer  with  two  Circles.  American 
Journal  of  Science,  1896  (4),  2,  298. 


CRYSTALLOGRAPHIC  INVESTIGATIONS  187 

ment,  as  it  really  requires  less  time  for  the  adjustment  of  the 
crystal. 

One  adjustment  of  a  crystal  in  true  polar  position  is  often 
sufficient  for  the  measurement  of  all  the  forms,  and  in  the 
case  of  colemanite,  less  than  an  hour  was  required  to  measure 
the  most  complex  combination. 

Polar  Orientation.  —  Every  face  of  a  crystal  has  its  position 
defined  with  reference  to  a  pole  and  a  direction  assumed  as 
first  meridian,  when  its  two  angular  distances,  respectively, 
from  these  are  known.  By  means  of  the  graduated  vertical 
and  horizontal  circles  of  the  instrument,  these  two  angular 
coordinates,  0  and  p,  are  readily  derived.  The  plane,  normal 
to  the  prismatic  zone,  is  preferably  chosen  as  the  pole-face, 
and  the  great  circle  passing  through  the  pole  and  the  normal 
to  the  side  pinacoid  (010)  as  the  first  meridian.  In  systems 
with  rectilinear  axes  the  pole-face  would  then  correspond  to  the 
basal-pinacoid,  and  the  angles  for  the  three  pinacoids  would  be 

<t>  P 

001  0°00  0°00 

010  0  00  90  00 

100  90  00  90  00 

Since  monoclinic  crystals  have  no  plane  normal  to  the  pris- 
matic zone,  the  position  of  such  a  plane  is  best  defined  if 
prismatic  faces  are  present.  Colemanite  possesses  a  well- 
developed  prismatic  zone,  including  both  pinacoids,  so  the 
polar  orientation  of  the  crystals  was  readily  accomplished. 
Each  face  of  a  crystal  comes  into  reflection  when  it  is  normal 
to  the  line  bisecting  the  angle  between  the  telescope  and 
collimator  of  the  instrument,  and  this  normal  position  must 
first  be  determined.  Its  angle  on  the  horizontal  circle  H, 
is  the  h0  reading  for  all  measurements.  The  method  of  find- 
ing ho  is  quite  simple.  Having  the  telescope  tightly  clamped 
at  a  convenient  distance  from  the  collimator,  a  bright 
reflecting  surface  is  then  mounted  approximately  parallel 
to  the  vertical  circle,  V,  and  centred.  It  is  then  brought  at 
the  intersection  of  the  crosshairs  by  turning  H  and  F,  and 
the  reading  on  H  taken,  =  hi.  V  is  then  turned  180°  and~&e 
reflection  again  brought  into  position  by  the  adjustment 
tables  and  H.  This  reading  on  H  =  hz.  Then  is  h0  = 


188  CRYSTALLOGRAPHY 


i  (/^  -^hz).  This  can  be  repeated  until  the  reflection  remains 
rigidly  fixed  at  the  intersection  of  the  crosshairs,  during  a 
revolution  of  V.  This  final  position  of  H  is  then  the  h0  and 
need  never  be  changed. 

The  crystal  of  colemanite  was  mounted  with  its  prismatic 
zone  approximately  normal  to  V,  and  H  was  clamped  at 
90°  +  h0.  The  reflection^from  the  prismatic  faces  were,  by 
means  of  the  centring  and  adjusting  tables,  brought  to 
revolve  directly  in  the  line  of  the  vertical  crosshair,  on  turn- 
ing V,  and  the  crystal  thus  brought  into  true  polar  position, 
because  a  plane  normal  to  the  prismatic  zone  would  then  be 
at  hQ.  The  reading  on  V  for  the  clinopinacoid  is  the  v0  for 
this  circle,  which  would  be  different  for  each  crystal.  Owing 
to  imperfect  centring  or  other  causes  readings  on  V  or  on 
H  for  certain  faces  vary,  when  they  should  be  the  same; 
consequently  both  VQ  and  hQ  can  be  corrected  by  averaging 
-  the  different  readings. 

When  the  reflection  of  each  face  is  brought  at  the  inter- 
section of  the  crosshairs,  two  readings,  one  on  V  =  v,  and 
one  on  H  =  h}  are  made;  the  face  is  then  defined  by  two 
angular  coordinates,  $  =  v  —  VQ  and  p  =  h  —  h0. 

Symbols  p  q.  —  In  place  of  the  three  indices  of  Miller  for 
a  terminal  form,  Goldschmidt  uses  the  two  indices  p  and  q, 
and  for  calculations  and  in  the  gnomonic  projection  the 
two  indices  are  preferable.  The  indices  of  Miller  are  readily 
transposed  into  those  of  Goldschmidt  by  making  the  last  one 
equal  to  unity  and  not  expressing  it;  thus  522  (Miller) 
becomes  fl  (Gdt).  Furthermore  001  =  0,  100  =  ooQ,  010  = 

0  oo     HO  =   co  ,   210  =  2  oo   =  ?  oo   120  =  -  oo  .      When   p  and 

q  p 

q  are  equal,  but  one  is  expressed,  thus  331  =  3.  The  zonal 
relations  of  forms  are  better  shown  by  the  two  symbols, 
because  all  forms  having  the  same  p,  or  the  same  q,  lie  in  the 
saine  straight  line  or  zone;  for  example,  it  can  be  seen  that  the 
forms  |f,  if,  |f  are  tautozonal,  whereas  their  Miller  equiva- 
lents (564),  (296),  (8.15.10)  do  not  show  this  relation  so  well. 
Gnomonic  Projection.  —  This  projection  shows  the  points 
of  Intersection  of  the  face-normals,  drawn  from  the  centre  of 
the  crystal,  upon  a  plane  lying  preferably  normal  to  the 
prismatic  zone  and  at  a  unit's  distance  from  the  centre  of  the 


CRYSTALLOGRAPHIC  INVESTIGATIONS 


189 


crystal.  If  h  is  the  distance  of  this  plane  above  the  centre 
and  is  equal  to  the  c-axis,  and  r0  is  the  length  of  the  base 
normal,  then  in  crystals  with  rectilinear  axes  r0  =  h  =  1. 
In  monoclinic  crystals  the  base  is  oblique  to  the  plane  of 

projection,    and    with    h  —  1,    r0    is   equal   to  -j  —  g  =  —  -  ; 

whence  it  follows  that  with  r0  =  1,'then  h  =  sin  ju. 

If  with  a  radius  of  h  =  1  =  c-axis,  a  circle  is  described, 
then  this  circle  would  represent  in  ground  plan  a  sphere  of 
projection.  The  plane  of  projection  would  be  tangent  to  this 
sphere  at  the  end  of  the  c-axis,  S  would  be  the  pole  of  the 
projection,  and  SY  the  first  meridian.  Any  face  pq  would 
have  the  point  of  intersection  of  its  normal  with  this  plane, 
located  by  the  angle  <j>  which  it  makes  with  the  first  meridian 
SY,  and  the  distance  from  the  pole  d'  =  tg  p.  (Figure  1). 
The  face  pq  is  further  defined  by  the  two  rectangular  coordi- 
nates xf  yf,  whose  values  deduced  from  the  right  triangles  are 


x    =  sin 
y'  =  cos 


tg  p 
tg  p 


By  means  of  the  measured  angles  <£  and  p,  all  forms  can  be 

plotted  on  cross  section  paper,  and  the  coordinates  x'  yf  give 

graphically  the  symbols  for  any  form  in  terms  of  the  x'  y' 

of  the  unit  form  pq.     In  monoclinic  crystals  the   projection 

of  the  base-normal  lies  in  front  of 

the  pole  at  a  distance  er  =  tg  p,  and 

the  distances  p'Q  and  q'Q  are  the  co- 

ordinates for   the   unit  pyramid    pq 

reckoned   from  the   base;    therefore 

for  any  values  of  pq,  x'  =  pp'0  +  e' 

and  y'  =  qq'Q. 

Determination  of  e',  /*,  and  |3.  —  An 
average  of  twenty  readings  on  the 
basal  pinacoid  gave  p  =  20°  7',  with 
</>  =  90°;  e'  =  tg  p  =  0.3663.  This 

value  of  e'  was  also  obtained  from  the  readings  of  the 
clinodomes.  For  these  forms  x'  =  e'  ',  and  an  average  of  forty- 
four  values  of  x'  for  the  domes  01  and  02  gave  x'  =  0.3663; 
thus,  agreeing  with  the  direct  measurements.  Also  e'  =  cot 
M  =  69°  53'  and  ft  =  180°  -  M  =  110°  7'. 


Fig.  1 


190 


CRYSTALLOGRAPHY 


Determination  of  p'Q  and  q'o-  —  By  means  of  the  two  formulae 


x'  =  sin 
y'  =  cos 


tg  p 
tg  p 


the  coordinates  xf  y'  were  calculated  for  all  the  best  faces. 
For  the  positive  forms  pp'o  =  x'  —  ef,  and  for  the  negative  pp'o 
=  x'  +  e'.  The  symbols  pq  are  simple  multiples  of  the  co- 
ordinates, p'0q'o  of  the  unit  form,  therefore  the  values  p'oq'o 
are  readily  deduced  for  each  form.  Taking  fifteen  of  the 
best  crystals  as  sufficient  for  the  calculation  of  the  elements, 
the  averages  of  p'o  and  q'o  were  as  follows: 


p'o  =  0.7431 

12  meas. 

7431 

9 

(i 

7441 

4 

it 

7425 

7 

(  ( 

7447 

17 

(( 

7452 

11 

it 

7452 

14 

u 

7444 

8 

(t 

7449 

10 

(I 

7435 

14 

11 

7450 

16 

11 

7447 

13 

11 

7453 

6 

11 

7446 

8 

(( 

7448 

16 

(( 

q'o  =  0.5438 

12  meas. 

5426 

9 

u 

5435 

9 

It 

5426 

7 

(( 

5426 

17 

11 

5435 

11 

u 

5432 

14 

it 

5429 

8 

11 

5417 

10 

11 

5424 

9 

u 

5436 

16 

u 

5423 

13 

11 

5437 

6 

It 

5425 

8 

u 

5431 

16 

(I 

Average  0.7443     165  meas.  0.5430       165  meas. 

The  elements  for  colemanite  are  therefore: 

p'o  =  0.7443;  q'0  =  0.5430;  ef  =  0.3663;  M  =  69°53'. 

Determination  of  the  Polar   Elements  p0,    q0  and  e.  —  The 
values    p'o,  qfo  and  e'  are  the  elements  when  h  =  1  and  r0  = 

^— ;   therefore  when  r0  =  1,  these  values  must  be  multiplied 

by  sin  ju,  to  obtain  the  polar  elements  p0,  qQ  and  e.  Thus 
po  =  P'O  sin  M,  #o  =  q'o  sin  ju  and  e  =  e'  sin  /*  =  cos  /*•  The 
elements  for  colemanite  then  become 

Po  =  0.6989;  q0  =  0.5098;  e  =  0.3439. 


CRYSTALLOGRAPHIC   INVESTIGATIONS  191 

Determination  of  the  Axial  Lengths  a  and  c.  —  In  systems 
with  rectilinear  axes  g'0  =  tg(001:  Oil)  and  p'0=  tg(001:  101); 
therefore  with  axis  6  =  1,  the  formulae  for  such  systems 

/        j    v        c         q'o 
become  c  =  q  o  and  a  =  —  -  =  -T- 

In   monoclinic   crystals,  c   is   also  equal  to  the  coordinate 

q'o;   the  clino-axis  a,  however,  is  equal  to  —  --  ,  therefore  the 

sin  ^ 

formulae  for   monoclinic   crystals,  in  terms   of   pQqQ  or  p'og'o, 
when  b  =  1,  become 


0  ~ 


sin 


From  these  equations,  a  =  0.7768  and  c  =  0.5430;  giving 
the  axial  ratio  for  colemanite  a:b:c  =  0.7768:1:0.5430. 

Record  of  the  Measurements.  —  As  an  illustration  of  the 
actual  measurements  and  method  of  deriving  the  symbols  for 
the  forms,  the  work  on  Crystal  23  is  given.  Each  crystal 
was  sketched  before  measuring  in  order  to  show  the  relative 
size  of  the  faces,  and  the  faces  were  provisionally  lettered. 
The  kind  of  reflection  was  designated  as  good,  fair,  or  poor 
(g.  f.  p.).  Starting  with  the  angle  on  the  side  pinacoid 
(010)  =  139°  44',  and  plotting  the  ^-angles  in  a  circle  in  the 
direction  of  the  hands  of  a  watch,  it  is  easily  determined 
which  forms  are  positive  and  which  negative.  The  h0  for 
the  instrument  was  57°  54',  and  the  v0  for  the  crystal  was 
139°  44'. 

From  the  columns  pp'o  and  qq'Q  the  symbols  pq  are  at 
once  apparent,  since  p'0  =  0.7443  and  q'0  =  0.5430.  The 
record  is  simply  a  sample  page  of  the  note  book,  and  it 
illustrates  the  simplicity  and  shortness  of  the  work  required 
to  determine  any  form,  as  well  as  the  neatness  and  compact- 
ness of  the  whole  method. 


192 


CRYSTALLOGRAPHY 


••3 

Letter. 

V 

h 

0  = 

I)  C0 

P  = 
h-ho 

Prisms  =  tg  <t> 

PP'o 
=  x'—e' 
(positive) 
=  x'+e' 
negative 

QQ'O 

Sym- 
bols 

P  Q 

Ig  sin  0 
Ig  tan  p 

Ig  COS   0 

lg*' 
igj/' 

x1 

y' 

g. 

6 

139°44 

147°54 

0°00 

90°00 

0 

Ooo 

g- 

m 

193  40 

147  54 

53  56 

90  00 

1.3730 

00 

00 

f. 

t 

209  27 

147  54 

69  43 

9000 

2.7058 

2oo 

00 

g. 

a 

229  44 

147  54 

90  00 

9000 

oo 

ooQ 

0 

g. 

f 

249  46 

147  54 

69  58 

9000 

2.7400 

2oo 

oo 

g. 

m! 

265  44 

147  54 

54  00 

90  00 

1.3755 

CO 

f. 

V 

319  44 

147  54 

000 

9000 

CO 

0 

0°o 

00 

f. 

t" 

35420 

147  54 

34  36 

9000 

0.6898 

oo2 

oo 

f. 

I 

345  07 

14754 

25  23 

90  00 

0.4745 

0=3 

CO 

g. 

m 

13  43 

14754 

53  59 

9000 

1.3752 

CO 

00 

0 

g. 

X 

229  43 

119  35 

89  59 

61  41 

0.268556 

0.268556 

1.8559 

1.4896 

0 

+20 

oo 

oo 

0 

0 

g. 

c 

229  43 

7801 

89  59 

2007 

9.563811 

9.563811 

0.3663 

0 

0 

0 

00 

oo 

0 

g. 

P 

203  42 

108  54 

63  58 

51  00 

9.953537 
0.091631 
9.642360 

0.045168 
9.733991 

1.1096 
0.5420 

0.7433 

0.5420 

+11 

g- 

g 

21755 

127  12 

78  11 

69  18 

9.990697 
0.422659 
9.311289 

0.413356 
9.733948 

2.5903 
0.5419 

2.2240 

0.5419 

+31 

g. 

g1 

241  30 

127  15 

78  14 

6921 

9.990777 
0.423807 
9.309474 

0.414584 
9.733281 

2.5977 
0.5411 

2.2314 

0.5411 

+31 

f. 

h 

243  55 

123  40 

75  49 

65  46 

9.986555 
0.346674 
9.389211 

0.333229 
9.735885 

2.1539 
0.5443 

1.7876 

0.5443 

+fl 

CRYSTALLOGRAPHIC  INVESTIGATIONS 


193 


a 

.2 

1 
1 

1 

V 

h 

0  = 

P  = 

h  ho 

Prisms  =  tg  0 

PP'o 

w'o 

Sym- 
bols 

P  Q 

Ig  sin  0 
Ig  tan  p 

Ig  COS  0 

Iga/ 

X 

y' 

—  x   e 
(positive) 

negative 

P- 

< 

261°51 

129°39 

57°53 

71°45 

9.927867 
0.481814 
9.725622 

0.409681 
0.207436 

2.5685 
1.6123 

2.2022 

1.6123 

+33 

g. 

P' 

255  45 

10857 

63  59 

51  03 

9.953599 
0.092406 
9.642101 

0.046005 
9.734507 

1.1117 
0.5427 

0.7454 

0.5427 

+11 

g- 

0 

158  24 

10642 

1840 

48  48 

9.505234 
0.055738 
9.976532 

9.560972 
0.032270 

0.3639 
1.0771 

0.0024 

1.0771 

02 

g- 

s 

17343 

91  05 

3359 

33  11 

9.747374 
9.815555 
9.918659 

9.562929 
9.734214 

0.3656 
0.5423 

0.0007 

0.5423 

01 

g- 

s' 

285  44 

91  06 

3400 

33  12 

9.747562 
9.815831 
9.918574 

9.563393 
9.734405 

0.3659 
0.5425 

0.0004 

0.5425 

01 

g. 

o' 

301  03 

10646 

18  41 

48  52 

9.505608 
0.058796 
9.976489 

9.564404 
0.035285 

0.3668 
1.0846 

0.0005 

1.0846 

02 

P- 

f 

28547 

12059 

33  57 

6305 

9.746999 
0.294397 
9.918830 

0.041396 
0.213227 

1.1000 
1.6339 

0.7337 

1.6339 

+  13 

g- 

V 

12028 

106  50 

19  16 

48  56 

9.518468 
0.059817 
9.974969 

9.578285 
0.034786 

0.3787 
1.0834 

0.7450 

1.0834 

-12 

g- 

w 

10447 

91  21 

34  57 

3327 

9.758050 
9.819959 
9.913630 

9.578009 
9.733589 

0.3785 
0.5415 

0.7448 

0.5415 

-1  1 

0 

g- 

n 

4945 

7839 

90  01 

2045 

9.578486 

9.578486 

0.3789 

0.7452 

0 

-10 

CO 

CO 

0 

194 


CRYSTALLOGRAPHY 


Re  ections. 

$ 

3 

V 

h 

0  = 

v  —  v0 

P  = 

h-h0 

Prisms  =  tg  <fe 

PP'o 
=  x'-e' 
(positive) 
=  x'  +  'e? 
[negative) 

vtfo 

Sym- 
bola 

P  Q. 

Ig  sin  <p 
Ig  tan  p 

Ig  COS  0 

Igx' 

igy' 

x' 

y' 

g- 

w' 

354°41 

91°24 

34°57 

33°30 

9.758050 
9.820783 
9.913630 

9.578833 
9.734413 

0.3792 
0.5425 

0.7455 

0.5425 

-1  1 

g- 

v' 

339  00 

106  53 

19  16 

48  59 

9.518468 
0.060582 
9.974969 

9.579050 
0.035551 

0.3794 
1.0853 

0.7457 

1.0853 

-12 

0 

g- 

d 

4945 

106  14 

9001 

4820 

0.050647 

0.050647 

1.1237 

1.4900 

0 

-20 

00 

CO 

0 

g- 

k 

23  58 

109  14 

64  14 

51  20 

9.954518 
0.096803 
9.638197 

0.051321 
9.735000 

1.1254 
0.5433 

1.4917 

0.5433 

-21 

g- 

u 

5  44 

115  18 

4600 

5724 

9.856934 
0.194141 
9.841771 

0.051075 
0.035912 

1.1247 

1.0862 

1.4910 

1.0862 

-22 

g. 

k' 

75  30 

109  11 

64  14 

51  17 

9.954518 
0.096027 
9.638197 

0.050545 
9.734224 

1.1234 
0.5423 

1.4897 

0.5423 

-21 

Calculated  Table.  —  Following  is  a  calculation  of  the  47 
forms  to  correspond  to  the  tables  arranged  by  Goldschmidt  in 
his  "  Krystallographische  Winkeltabellen."  The  calculations 
of  colemanite  given  by  the  writer  are  based  on  the  elements 
po  =  0.6989;  q0  =  0.5098;  and  e  =  0.3439,  and  therefore 
show  a  slight  difference  in  the  angles  from  those  given  by 
him  on  page  100. 


CRYSTALLOGRAPHIC  INVESTIGATIONS 


195 


a  =  .7768 

lg  a  =  9.890309          lg  ao  =  0.155509 

lgpo  =  9.844415      ao=1.4306  po  =  .6989 

c  =  .5430 

lg  c  =  9.734800           lg  bo  =  0.265200 

lg  go  =  9.707463       60  =  1.8416  <?o  =  .  5098 

?»•-/»  I6™ 

lg  riTj*  |  9'972663     lg  cos  p  \  9.536474 

lg  g  =  0.136952      h  =0.9390     e=  .3439 

No. 

1  Letter. 

Gdt. 

Miller. 

0 

P 

to 

. 

.« 

' 

x' 

(Prisms) 
(x  :  y) 

y' 

d' 

1 

C 

0 

001 

90°00 

20°07 

20°07 

0°00 

20°07 

0°00 

0.3663 

0 

.3663 

2 

b 

Ooo 

010 

0°00 

90°00 

0°00 

90°00 

0°00 

90°00 

0 

CO 

00 

3 

a 

ooQ 

100 

90°00 

90°00 

90°00 

0°00 

90°00 

0°00 

CO 

0 

ft 

4 

I 

3oo 

310 

76°20 

it 

it 

90°00 

76°20 

13°40 

4.1227 

GO 

" 

5 

t 

2oo 

210 

69°58 

u 

ft 

" 

69°58 

20°02 

2.7419 

" 

tt 

6 

m 

00 

110 

53°53' 

ii 

tt 

" 

53°53' 

36°06' 

1.3709 

" 

" 

7 

z 

oo2 

120 

34°26 

it 

" 

" 

34°26 

55°34 

0.6854 

tt 

It 

8 

H 

oo3 

130 

24°33' 

" 

" 

" 

24°33' 

65°26' 

0.4570 

« 

" 

9 

K 

01 

Oil 

34°00 

33°13* 

20°07 

28°30 

17°50' 

27°01 

0.3663 

0.5430 

0.6550 

10 

a 

02 

021 

18°38 

48°54 

a 

47°22 

13°56 

45°34 

it 

1.0860 

1.1461 

11 

V 

+  10 

101 

90°00 

47°59 

48°00 

0°00 

47°59 

0°00 

1.1106 

0 

1.1106 

12 

X 

+  20 

201 

11 

61°40 

61°40 

it 

61°40 

« 

1.8549 

" 

1.8549 

13 

p 

+  30 

301 

tt 

68°57' 

68°57' 

" 

68°57' 

tt 

2.5992 

" 

2.5992 

14 

i 

-  10 

101 

9b°00 

20°42' 

20°42' 

u 

20°42 

d 

0.3781 

n 

0.3781 

15 

h 

-20 

201 

(t 

48°18 

48°18 

1C 

48°18 

« 

1.1223 

tt 

1.1223 

16 

g 

-to 

502 

It 

56°13 

56°13 

« 

56°13 

a 

1.4944 

« 

1.4944 

17 

w 

-  30 

301 

u 

61°49 

61°49 

it 

61°49 

it 

1.8666 

a 

1.8666 

18 

t 

-40 

401 

tt 

69°02 

69°02 

tt 

69°02 

u 

2.6109 

" 

2.6109 

19 

u 

-  60 

601 

it 

76°17' 

7b°17 

" 

76°17 

(I 

4.0996 

u 

4.0996 

20 

f 

-  80 

801 

K 

79°51 

79°51 

" 

79°51 

tt 

5.5882 

u 

5.5882 

21 

a 

+  3 

331 

57°55 

71°56' 

68°57 

58°27' 

53°40 

30°19' 

2.5992 

1.6290 

3.0675 

22 

ft 

+  1 

111 

63°56' 

51°02 

48°00 

28°30 

44°18 

19°58 

1.1106 

0.5430 

1.2362 

23 

y 

-  1 

111 

34°51 

33°29' 

20°42 

" 

18°23 

26°55' 

0.3781 

" 

0.6617 

24 

V 

-2 

221 

45°56' 

57°22 

48°18 

47°22 

37°14' 

35°51 

1.1223 

1.0860 

1.5617 

25 

q 

Q 

331 

48°53 

68°01 

61°49 

58°27' 

44°19 

37°34 

1.8666 

1.6290 

2.4776 

26 

n 

+  14 

141 

27°05 

67°42 

48°00 

65°17 

24°54' 

55°28 

1.1106 

2.1720 

2.4396 

27 

CO 

+  13 

131 

34°17 

63°06' 

tt 

58°27' 

30°09 

47°28 

tt 

1.6290 

1.9716 

28 

e 

+  12 

121 

45°38 

57°13' 

48°00 

47°22 

36°57 

36°00' 

1.1106 

1.0860 

1.5533 

29 

77 

+  Is 

232 

53°44' 

54°01 

u 

39°10 

40°44 

28°35' 

a 

0.8145 

1.3772 

30 

r 

-If 

232 

24°54 

41°55' 

20°42 

« 

16°20 

37°18 

0.3781 

1.1100 

0.8980 

31 

d 

-  12 

121 

19°ir 

48°59* 

u 

47°22 

14°22 

45°27' 

d 

1.0860 

1.1500 

32 

X 

-  13 

131 

13°04 

59°07 

(I 

58°27' 

ll°ll 

56°43' 

" 

1.6290 

1.6724 

33 

C 

+  10.1 

10.1.1 

86°or 

82°43 

82°42' 

28°30 

81°42' 

3°57 

7.8093 

0.5430 

7.8281 

34 

k 

+  31 

311 

78°12 

69°22 

68°57' 

it 

66°21 

11°02 

2.5992 

tt 

2.6553 

35 

<t> 

+  tl 

522 

76°18 

66°26 

65°49 

« 

62°56 

12°32' 

2.2270 

tt 

2.2922 

36 

0 

-  21 

211 

64°11 

51°16 

48°18 

" 

44°36' 

19°52 

1.1223 

tt 

1.2468 

37 

© 

-31 

311 

73°47 

62°47 

6T°49 

" 

58°38 

14°23 

1.8666 

n 

1.9441 

38 

B 

-41 

411 

78°15 

69°26 

69°02' 

" 

66°27 

10°59 

2.6109 

n 

2.6663 

39 

t 

-23 

231 

34°34 

63°ir 

48°18 

57°27. 

30°25 

47°18 

1.1223 

1.6290 

1.9782 

196 


CRYSTALLOGRAPHY 


o  =  .7768 

lg  a  =  9.890309 

lg  oo  =  0.155509 

lgpo  =  9.8444  15 

ao=1.4306 

po  =  .6989 

c  =  .5430 

lgc  =  9.734800 

lg  bo  =  0.265200 

lg  50  =  9.707463 

feo=  1.8416 

90  =.5098 

M  = 
180°  -/3 

69°  53' 

le  A=     ) 
„  >  9.972663 
lgsm/A  J 

!gc=,,  [  9.536474 
IgcosM  ) 

lg  —  =  0.136952 
Qo 

h  =  0.93  90 

e=.3439 

No. 

1 

Gdt. 

Miller. 

0 

P 

lo 

170 

t 

V 

x' 
(Prisms) 
(x:y) 

y' 

a" 
=  tgP 

40 

Q 

-24 

241 

27°19* 

67°45 

48°18' 

65°17 

25°08. 

55°19 

T.1223 

2.1720 

2.4450 

41 

s 

-  34 

341 

40°4Cr 

70°45 

61°49 

a 

37°58' 

45°43' 

1.8666 

u 

2.8640 

42 

7 

-32 

321 

59°48' 

65°09 

u 

47°22 

51°39' 

27°09 

u 

1.0860 

2.1598 

43 

w 

-H 

182 

0°09' 

65°17 

0°20 

65°17 

0°08' 

65°17 

0.0058 

2.1720 

2.1715 

44 

p 

+  *2 

142 

34°12' 

52° 

43 

36°26' 

47°22 

26°34 

41°08' 

0.7384 

1.0860 

1.3133 

45 

u 

+  H 

164 

34°08' 

44°32 

28°55 

39°10 

23°11 

35°29' 

0.5523 

0.8145 

0.9841 

46 

/* 

+  H 

165 

38°19' 

39° 

43. 

27°15 

33°05° 

23°20. 

30°05 

0.5151 

0.6516 

0.8306 

47 

p 

-H 

123 

18°05 

20°51 

6°44 

19°54 

6°20. 

19°46' 

0.1182 

0.3620 

0.3808 

It  is  to  be  regretted  that  up  to  the  present  no  descriptive 
articles  dealing  with  crystals  of  the  tetragonal  and  triclinic 
systems  according  to  the  two-circle  method  have  yet  appeared 
in  the  English  language.  Very  good  illustrations  of  the  applica- 
tion of  this  method  to  crystals  of  these  systems  have  been 
published  in  German,  viz.: 


TETRAGONAL  SYSTEM 

tlber  Kalomel  von  Victor  Goldschmidt  und  B.  Mauritz 
(Zeitschrift  fur  Krystallographie,  Band  XLIV). 

TRICLINIC  SYSTEM 

Krystallberechnung  im  triklinen  System  illustriert  am 
Anorthit  von  L.  Borgstrom  und  Victor  Goldschmidt  (Zeit- 
schrift fur  Krystallographie,  Band  XLI). 


CRYSTALLOGRAPHIC    INVESTIGATIONS 


197 


Plate  I 


198 


CRYSTALLOGRAPHY 


Colemanite 
Plate  2 


CRYSTALLOGRAPHIC    INVESTIGATIONS 
7  8 


199 


Colemanite 
Plate  3 


INDEX 


Acumination,  33 

Aggregates,  fibrous,  radiating,  144 

Alum,  6 

Amorphous  substances,  44 

Analogue  pole,  124 

Antilogue  pole,  124 

Apatite  from  Minot,  Maine,  171 

Axes  of  reference,  48,  71,  90,  115,  126, 

137 

Axes  of  symmetry,  37 
Axial  lengths,  determination  of,  191 
Axial  ratios,  31,  77,  96,  119,  131,  138 
Axis,  twinning,  146 

Basal  edges,  93 

Baveno  twinning  law,  152 

Bergmann,  44 

Bevelment,  33 

Brazilian  law,  151 

Butterfly  twin  of  calcite,  151 

Calculations,  record  of,  27 
Carangeot,  18 

Central  distances  of  faces,  36 
Chemical  properties  of  crystals,  6 
Circular  polarization,  81 
Clinographic  drawings,  160,  162 
Closed  forms,  76 
Combinations,  39 
Combinations,  cubic,  54 
Combinations,  hexagonal,  95 
Combinations,  monoclinic,  130 
Combinations,  rhombic,  121 
Combinations,  tetragonal,  78 
Combinations,  triclinic,  139 
Combination  of  crystal  forms,  39,  54 
Composition  face,  146 
Congruent  forms,  61 
Constancy  of  interfacial  angles,  15 


Contact  twins,  146 

Colemanite  from  Southern  California, 

177 
Co-ordinate    representation    of   face 

direction,  29 

Corpuscular  hypothesis,  46 
Crystals,  chemical  properties,  6 
Crystals,  formation  of,  3 
Crystals,  pyro-electric,  87 
Crystals,   the  internal  structure   of, 

43 

Crystal  aggregates,  143 
Crystal  classes,  38 
Crystal  defined,  1 
Crystal  drawing,  160 
Crystal  forms,  14,  36 
Crystal  surfaces,  irregularities,  156 
Crystal  systems,  39 
Crystal  symbols,  13 
Crystalline  condition,  46 
Crystallized  condition,  46 
Crystallographic  constants,  25 
Cube,  symmetry  of,  38 
Cubic  system,  47 
Cubic  system,  twins,  148 
Curved  faces,  156 
Cyclic  twinning,  148 

Dauphine  law,  151 
Dimorphism,  9 
Distortion,  36 
Double  refraction,  42 
Drawing  of  crystals,  160 

Eakle,  177 

Enantiomorphous  forms,  57,  67,  105 
Etching  figures,  62,  158 
Equivalent  axes,  71,  90 


202 


INDEX 


Formation  of  crystals,  2 

Formation  of  crystals  by  sublimation, 

4 
Formation  of  crystals  from  solution, 

3 
Formation  of  crystals  from  fusion,  4 

General  formula,  7 

General  symbols,  52 

Geniculate  twin,  149 

Goethite,  notes  on,  176 

Gnomonic  projection,  10,  22,  68,  88, 

97,  119,  131,  141,  188,  197 
Goldschmidt,  21,  25,  163,  165,  176 
Goldschmidt  symbols,  13 
Goniometers,  17,  18,  19,  21 
Goniometer,    two-circle,    method    of 

measurement,  177 
Ground  circle,  120 
Ground  form,  73,  115 

Haiiy,  46 

Hemihedral  classes,  cubic,  56 
Hemihedral  class,  monoclinic,  133 
Hemihedral  class,  triclinic,  141 
Hemihedral    hemimorphic    class,    te- 
tragonal, 87 

Hemimorphic  axis,  86,  97,  124 
Hemimorphic  class,  monoclinic,  133 
Hemimorphic  class,  rhombic,  123 
Hemimorphic  forms,  86 
Hemimorphism,  86 
Hexagonal  symbols,  91 
Hexagonal  system,  89 
Hexagonal  system,  twins,  150 
Holohedral  class,  cubic,  47 
Holohedral  class,  hexagonal,  93 
Holohedral  class,  monoclinic,  125 
Holohedral  class,  rhombic,  114 
Holohedral  class,  tetragonal,  70 
Holohedral  class,  triclinic,  136 
Holohedral  hemimorphic  class,  hex- 
agonal, 97 

Holohedral    hemimorphic    class,    te- 
tragonal, 86 

Idealized  crystals,  37 
Icosihedron,  63 


Indices,  13,  31 
Inequivalent  axes,  115 
Interfacial  angles,  15,  18 
Interpenetration  twins,  147 
Internal  structure  of  crystals,  43,  45 
Irregular  aggregates,  144 
Iron  cross  twin  of  pyrite,  149 
Isomorphism,  7 
Isomorphous  compounds,  7 
Isomorphous  elements,  7 
Isomorphous  substances,  7 

Karlsbad  twinning  law,  152 

Lamellar  twinning,  147 

Law  of  constancy  of  interfacial  angles, 

15 

Law  of    rationality  of  axial   param- 
eters, 13 

Law  of  symmetry,  15 
Leit  Linie,  163 
Limiting  forms,  cubic,  55 
Limiting  forms,  hexagonal,  96 
Limiting  forms,  monoclinic,  130 
Limiting  forms,  rhombic,  118 
Limiting  forms,  tetragonal,  78 
Limiting  forms,  triclinic,  140 

Mallard,  154 

Mannebach  twinning  law,  152 

Mathematical  characteristics,  10 

Measurement  of  crystals,  16 

Measurements,  form  of  record,  191 

Mica  twinning  law,  153 

Miers,  58 

Miller's  index  symbols,  13,  31 

Mimicry,  152 

Minimum  cohesion,  41 

Mitscherlich,  6 

Monoclinic  system,  125 

Monoclinic  system,  twins,  152 

Nicol,  165 
Normal  model,  10 

Open  forms,  75,  95,  96,  117 
Optic  axis,  42 
Orthographic  drawings,  160 


INDEX 


203 


Orthorhombic  system,  (See  Rhombic) 

Palache,  171 

Parallel  growths,  143 

Parameters,  32 

Parsons,  176 

Pasteur,  135 

Penetration  twins,  147 

Penfield,  159 

Pentagonal  hemihedral  class,  59 

Physical  properties,  41 

Plagiohedral  hemihedral  class,  57 

Plan,  preparation  from  a  gnomonic 
projection,  161 

Planes  of  symmetry,  35 

Polar  edges,  93 

Polar  elements,  16,  31,  77,  97,  120, 
132,  141 

Polyhedra,  general  properties,  32 

Polysynthetic  twinning,  147 

Projections,  10,  22,  28,  87,  97,  119, 
131,  140 

Projections  of  galena,  69 

Pseudo-symmetry,  155 

Purity  of  crystals,  43 

Pyramidal  hemihedral  class,  hexag- 
onal, 102 

Pyramidal   hemihedral   class,  tetrag- 
onal, 81 

Pyramidal  hemihedral  hemimorphic 
class,  104 

Pyritohedron,  39 
Pyro-electric  crystals,    124 

Rationality  of  the  pace  numbers,  13 
Rationality  of  the  axial  parameters, 

13 

Record  of  measurements,  191 
Refraction  of  light,  41 
Repeated  twinning,  147 
Rhombic  system,  113 
Rhombic  system,  twins,  151 
Rhombohedral  hemihedral  class,  98 
Rhombohedral    tetratohedral    class, 

107 
Rhombohedron  of  the  middle  edges, 

100 


Single  refraction,  41 
Sperrylite,  new  forms,  165 
Sphenoidal  class,  rhombic,  121 
Sphenoidal  hemihedral  class,  extrag- 

onal,  83  v 

Spinel  law,  148 

Standard  orientation,  cubic,  47 
Standard  orientation,  hexagonal,  89 
Standard  orientation,  rhombic,  114 
Standard  orientation,  monoclinic,  125 
Standard  orientation,  tetragonal,  71 
Standard  orientation,  triclinic,  136 
Steno,  Nicholas,  15 
Stereographic  projections,  28,  69,  88, 

97 

Stereoisomers,  135 
Striations  due  to  twinning,  153 
Striations  on  crystals,  62,  158 
Supplementary  twinning,  149 
Swallow-tailed  twin,  144,  146 
Symbols,  13,  25,  31,  32 
Symmetry,  70,  125 
Symmetry,  law  of,  15 
Symmetry  of  crystals,  35 
Symmetry  of  the  cubic  system,  47 
Systems  of  crystals,  40 

Tetragonal  system,  70 
Tetragonal  system  twins,  147 
Tetrahedral  forms,  66 
Tetrahedral    hemihedral    class,    63 
Tetartohedral  class,  cubic,  66 
Tetartohedral  class,  tetragonal,  85 
Tetartohedrism,  evidence  of,  69 
Topaz  projection,  11 
Tourmaline,  pyro-electric,  112 
Trapezohedral  hemihedral  class,  hex- 
agonal, 101 
Trapezohedral       hemihedral      class, 

tetragonal,  79 
Trapezohedral     tetartohedral     class, 

hexagonal,  104 
Triclinic  system,  136 
Triclinic  system  twins,  153 
Trigonal  hemihedral  class,  109 
Trigonal     hemihedral     hemiraorphic 

class,  111 
Trigonal  tetartohedral  class,  112 


204  INDEX 

Trigonal  tetartohedral  hemimorphic      Unit  pyramid,  73 

class,  113  Vicinal  planes,  157 

Trimorphism,  9  Weigg,  symboiS)  32)  50 

Truncation,  33  winkel  punkt>  163 

Twinning,  144  Wolff>  171 

Twinning  axis,  146  Wollaston,  18 
Twinning,  chief  types,  146 

Twinning,  general  effects,  154  Zone,  12 

Twinning  plane,  145  Zone  lines,  12 


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